%------------------------------------------------------------------------------
% File     : ITP278^3 : TPTP v9.0.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00535_035038
% Version  : [Des22] axioms.
% English  :

% Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
%          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% Source   : [Des22]
% Names    : 0075_VEBT_Uniqueness_00535_035038 [Des22]

% Status   : Theorem
% Rating   : 0.75 v9.0.0, 0.50 v8.2.0, 0.46 v8.1.0
% Syntax   : Number of formulae    : 11471 (5791 unt;1229 typ;   0 def)
%            Number of atoms       : 29163 (12297 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 124413 (2951   ~; 455   |;1841   &;108012   @)
%                                         (   0 <=>;11154  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  102 ( 101 usr)
%            Number of type conns  : 5085 (5085   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1131 (1128 usr;  78 con; 0-8 aty)
%            Number of variables   : 27013 (2586   ^;23805   !; 622   ?;27013   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 16:35:31.415
%------------------------------------------------------------------------------
% Could-be-implicit typings (101)
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% Explicit typings (1128)
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    bNF_re728719798268516973at_o_o: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( ( nat > rat ) > $o ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_Eo_001_Eo,type,
    bNF_re4297313714947099218al_o_o: ( ( nat > rat ) > real > $o ) > ( $o > $o > $o ) > ( ( nat > rat ) > $o ) > ( real > $o ) > $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_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_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_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__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__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_BNF__Wellorder__Constructions_OordIso_001_Eo_001_Eo,type,
    bNF_We2654380646378065620so_o_o: set_Pr1932065953672099015od_o_o ).

thf(sy_c_BNF__Wellorder__Constructions_OordIso_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_BNF__Wellorder__Constructions_OordLess_001_Eo_001t__Nat__Onat,type,
    bNF_We8182288985678559134_o_nat: set_Pr457366540195662369at_nat ).

thf(sy_c_BNF__Wellorder__Relation_Owo__rel_001t__Nat__Onat,type,
    bNF_We3818239936649020644el_nat: set_Pr1261947904930325089at_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__Code____Numeral__Ointeger,type,
    bit_se3949692690581998587nteger: code_integer > code_integer > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se1080825931792720795nteger: code_integer > code_integer > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

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__Code____Numeral__Ointeger,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

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__class_Oand__num,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
    bit_un4731106466462545111um_rel: product_prod_num_num > product_prod_num_num > $o ).

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

thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
    bit_un2901131394128224187um_rel: product_prod_num_num > product_prod_num_num > $o ).

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

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

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

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

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

thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
    code_integer_of_nat: nat > code_integer ).

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

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

thf(sy_c_Code__Numeral_Onegative,type,
    code_negative: num > code_integer ).

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

thf(sy_c_Code__Numeral_Opositive,type,
    code_positive: num > code_integer ).

thf(sy_c_Code__Target__Int_Onegative,type,
    code_Target_negative: num > int ).

thf(sy_c_Code__Target__Int_Opositive,type,
    code_Target_positive: num > int ).

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

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

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

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

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

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

thf(sy_c_Complex_Ocomplex_OComplex,type,
    complex2: real > real > complex ).

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

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

thf(sy_c_Complex_Ocsqrt,type,
    csqrt: complex > complex ).

thf(sy_c_Complex_Oimaginary__unit,type,
    imaginary_unit: complex ).

thf(sy_c_Complex_Orcis,type,
    rcis: real > real > complex ).

thf(sy_c_Deriv_Ohas__field__derivative_001t__Real__Oreal,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(sy_c_Divides_Oadjust__div,type,
    adjust_div: product_prod_int_int > int ).

thf(sy_c_Divides_Oadjust__mod,type,
    adjust_mod: int > int > int ).

thf(sy_c_Divides_Odivmod__nat,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Divides_Oeucl__rel__int,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Code____Numeral__Ointeger,type,
    unique5706413561485394159nteger: produc8923325533196201883nteger > $o ).

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux_001t__Int__Oint,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

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

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

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

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

thf(sy_c_Extended__Nat_OeSuc,type,
    extended_eSuc: extended_enat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat,type,
    extended_enat2: nat > extended_enat ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
    extended_case_enat_o: ( nat > $o ) > $o > extended_enat > $o ).

thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity_001t__Extended____Nat__Oenat,type,
    extend5688581933313929465d_enat: extended_enat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Complex__Ocomplex,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Nat__Onat,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
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thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Complex__Ocomplex,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
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thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
    inverse_inverse_real: real > real ).

thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
    at_bot_real: filter_real ).

thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
    at_top_nat: filter_nat ).

thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofinite__subsets__at__top_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ocard_001_Eo,type,
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thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Rat__Orat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Fun_Obij__betw_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_Fun_Obij__betw_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Ocomp_001_062_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Fun_Oid_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_Fun_Oinj__on_001t__Set__Oset_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Groups_Oabs__class_Oabs_001t__Real__Oreal,type,
    abs_abs_real: real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_I_Eo_M_Eo_J,type,
    minus_minus_o_o: ( $o > $o ) > ( $o > $o ) > $o > $o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Int__Oint_M_Eo_J,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Real__Oreal_M_Eo_J,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
    minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex,type,
    minus_minus_complex: complex > complex > complex ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
    minus_minus_int: int > int > int ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Rat__Orat,type,
    minus_minus_rat: rat > rat > rat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Real__Oreal,type,
    minus_minus_real: real > real > real ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_Eo_J,type,
    minus_minus_set_o: set_o > set_o > set_o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

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

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

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > rat > rat ).

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

thf(sy_c_Groups_Oplus__class_Oplus_001t__String__Oliteral,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Int__Oint,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Rat__Orat,type,
    sgn_sgn_rat: rat > rat ).

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

thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
    times_times_int: int > int > int ).

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

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

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

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_I_Eo_M_Eo_J,type,
    uminus_uminus_o_o: ( $o > $o ) > $o > $o ).

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

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
    uminus_uminus_rat: rat > rat ).

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

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    uminus8566677241136511917omplex: set_complex > set_complex ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Int__Oint_J,type,
    sup_sup_set_int: set_int > set_int > set_int ).

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

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

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Num__Onum_J,type,
    sup_sup_set_num: set_num > set_num > set_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    sup_su6024340866399070445nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    sup_su6327502436637775413at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    sup_su2042722026077122175at_num: set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num > set_Pr6200539531224447659at_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    sup_su4061117120043295689um_num: set_Pr8218934625190621173um_num > set_Pr8218934625190621173um_num > set_Pr8218934625190621173um_num ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Rat__Orat_J,type,
    sup_sup_set_rat: set_rat > set_rat > set_rat ).

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

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

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

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

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

thf(sy_c_List_Oappend_001t__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_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_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    map_VE8901447254227204932T_VEBT: ( vEBT_VEBT > vEBT_VEBT ) > list_VEBT_VEBT > list_VEBT_VEBT ).

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

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

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

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

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_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_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

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

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

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

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

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_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__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

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

thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    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__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__Num__Onum,type,
    product_nat_num: list_nat > list_num > list_P1726324292696863441at_num ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_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__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_Otake_001t__VEBT____Definitions__OVEBT,type,
    take_VEBT_VEBT: nat > list_VEBT_VEBT > list_VEBT_VEBT ).

thf(sy_c_List_Ounion_001t__Nat__Onat,type,
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thf(sy_c_List_Ounion_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    union_4462254032241401953at_nat: list_P5464809261938338413at_nat > list_P5464809261938338413at_nat > list_P5464809261938338413at_nat ).

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

thf(sy_c_List_Oupt,type,
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thf(sy_c_List_Oupto,type,
    upto: int > 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,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
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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,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat__Bijection_Olist__decode,type,
    nat_list_decode: nat > list_nat ).

thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
    nat_list_decode_rel: nat > nat > $o ).

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,type,
    nat_prod_decode: nat > product_prod_nat_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

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

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

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

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

thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

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

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

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

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

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

thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Order__Relation_Owell__order__on_001t__Nat__Onat,type,
    order_2888998067076097458on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    bot_bot_complex_o: complex > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    bot_bot_int_int_o: int > int > $o ).

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

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    bot_bot_nat_nat_o: nat > nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    bot_bot_nat_num_o: nat > num > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J,type,
    bot_bot_num_num_o: num > num > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
    bot_bot_real_o: real > $o ).

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

thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    bot_bo394778441745866138_nat_o: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
    bot_bot_o: $o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
    bot_bot_filter_nat: filter_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_Eo_J,type,
    bot_bot_set_o: set_o ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    bot_bot_set_complex: set_complex ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
    bot_bot_set_num: set_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
    bot_bo7038385379056416535at_num: set_Pr6200539531224447659at_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
    bot_bo9056780473022590049um_num: set_Pr8218934625190621173um_num ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
    bot_bot_set_rat: set_rat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
    bot_bot_set_real: set_real ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat,type,
    ord_Least_nat: ( nat > $o ) > nat ).

thf(sy_c_Orderings_Oord__class_Oless_001_062_I_Eo_M_Eo_J,type,
    ord_less_o_o: ( $o > $o ) > ( $o > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

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

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

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_I_Eo_M_Eo_J,type,
    ord_less_eq_o_o: ( $o > $o ) > ( $o > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_062_It__Num__Onum_M_Eo_J_J,type,
    ord_le3404735783095501756_num_o: ( nat > num > $o ) > ( nat > num > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Num__Onum_M_062_It__Num__Onum_M_Eo_J_J,type,
    ord_le6124364862034508274_num_o: ( num > num > $o ) > ( num > num > $o ) > $o ).

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

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    ord_le3935385432712749774_nat_o: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ) > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001_Eo,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Extended____Nat__Oenat,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Rat__Orat,type,
    ord_less_eq_rat: rat > rat > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Code____Numeral__Ointeger_J,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
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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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_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,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
    ord_max_nat: nat > nat > nat ).

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
    ord_min_nat: nat > nat > nat ).

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

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,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_Oordering__top_001t__Nat__Onat,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__List__Olist_It__Nat__Onat_J_J,type,
    top_top_set_list_nat: set_list_nat ).

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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_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__Code____Numeral__Ointeger,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,
    power_power_rat: rat > nat > rat ).

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_M_Eo_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__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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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,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Num__Onum_Mt__Num__Onum_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J_J,type,
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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,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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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_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_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__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__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__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc2761476792215241774st_nat: ( nat > nat > list_nat ) > product_prod_nat_nat > list_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
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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,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_I_Eo_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Complex__Ocomplex_J,type,
    produc2866383454006189126omplex: ( num > num > set_complex ) > product_prod_num_num > set_complex ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Int__Oint_J,type,
    produc6406642877701697732et_int: ( num > num > set_int ) > product_prod_num_num > set_int ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Num__Onum_001t__Num__Onum_001t__Set__Oset_It__Real__Oreal_J,type,
    produc8296048397933160132t_real: ( num > num > set_real ) > product_prod_num_num > set_real ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001_Eo,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
    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_Product__Type_Ounit_OAbs__unit,type,
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thf(sy_c_Product__Type_Ounit_ORep__unit,type,
    product_Rep_unit: product_unit > $o ).

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,
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thf(sy_c_Rat_Onormalize,type,
    normalize: product_prod_int_int > product_prod_int_int ).

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_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_OReal,type,
    real2: ( nat > rat ) > real ).

thf(sy_c_Real_Ocauchy,type,
    cauchy: ( nat > rat ) > $o ).

thf(sy_c_Real_Opcr__real,type,
    pcr_real: ( nat > rat ) > real > $o ).

thf(sy_c_Real_Opositive,type,
    positive: real > $o ).

thf(sy_c_Real_Orealrel,type,
    realrel: ( nat > rat ) > ( nat > rat ) > $o ).

thf(sy_c_Real_Orep__real,type,
    rep_real: real > nat > rat ).

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_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_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: real > real ).

thf(sy_c_Relation_OField_001_Eo,type,
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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_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

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

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

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

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

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

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

thf(sy_c_Rings_Odvd__class_Odvd_001t__Real__Oreal,type,
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thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Nat__Onat,type,
    modulo_modulo_nat: nat > nat > nat ).

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thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Int__Oint,type,
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    zero_n2687167440665602831ol_nat: $o > nat ).

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    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,
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thf(sy_c_Series_Osuminf_001t__Nat__Onat,type,
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thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
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thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
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thf(sy_c_Series_Osummable_001t__Nat__Onat,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
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thf(sy_c_Series_Osums_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Ochar,type,
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thf(sy_c_Set_Oimage_001t__String__Ochar_001t__Nat__Onat,type,
    image_char_nat: ( char > nat ) > set_char > set_nat ).

thf(sy_c_Set_Oinsert_001_Eo,type,
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thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
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thf(sy_c_Set_Oinsert_001t__List__Olist_It__Nat__Onat_J,type,
    insert_list_nat: list_nat > set_list_nat > set_list_nat ).

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

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

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

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

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

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

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

thf(sy_c_Set_Ois__singleton_001_Eo,type,
    is_singleton_o: set_o > $o ).

thf(sy_c_Set_Ois__singleton_001t__Complex__Ocomplex,type,
    is_singleton_complex: set_complex > $o ).

thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
    is_singleton_int: set_int > $o ).

thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
    is_singleton_nat: set_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    is_sin2850979758926227957at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    is_sin2937591304547752795at_nat: set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_Set_Ois__singleton_001t__Real__Oreal,type,
    is_singleton_real: set_real > $o ).

thf(sy_c_Set_Othe__elem_001_Eo,type,
    the_elem_o: set_o > $o ).

thf(sy_c_Set_Othe__elem_001t__Int__Oint,type,
    the_elem_int: set_int > int ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001_Eo,type,
    set_or8904488021354931149Most_o: $o > $o > set_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__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

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

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001_Eo,type,
    set_ord_atMost_o: $o > set_o ).

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__Nat__Onat_J,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

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__Nat__Onat,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001_Eo,type,
    set_ord_lessThan_o: $o > set_o ).

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_OCode_Oabort_001t__Real__Oreal,type,
    abort_real: literal > ( product_unit > real ) > real ).

thf(sy_c_String_OLiteral,type,
    literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).

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_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__Nat__Onat_J,type,
    topolo7278393974255667507et_nat: ( nat > set_nat ) > $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_Oconvergent_001t__Real__Oreal,type,
    topolo7531315842566124627t_real: ( nat > real ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_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_Opowr__real,type,
    powr_real2: 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_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

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

thf(sy_c_Typedef_Otype__definition_001t__Product____Type__Ounit_001_Eo,type,
    type_d6188575255521822967unit_o: ( product_unit > $o ) > ( $o > product_unit ) > set_o > $o ).

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
    vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).

thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
    vEBT_V312737461966249ad_rel: produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Extended____Nat__Oenat_J,type,
    accp_P6183159247885693666d_enat: ( produc7272778201969148633d_enat > produc7272778201969148633d_enat > $o ) > produc7272778201969148633d_enat > $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_Oless__than,type,
    less_than: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Omax__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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__Int__Oint_J,type,
    member_list_int: list_int > set_list_int > $o ).

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

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

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

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
    member9148766508732265716at_num: product_prod_nat_num > set_Pr6200539531224447659at_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
    member7279096912039735102um_num: product_prod_num_num > set_Pr8218934625190621173um_num > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_J,type,
    member444158400953824016od_o_o: produc2934264451710624999od_o_o > set_Pr1932065953672099015od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member4095101504841534314at_nat: produc732395585841259969at_nat > set_Pr457366540195662369at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

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

thf(sy_v_deg____,type,
    deg: nat ).

thf(sy_v_info____,type,
    info: option4927543243414619207at_nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_sa____,type,
    sa: vEBT_VEBT ).

thf(sy_v_summary_H____,type,
    summary: vEBT_VEBT ).

thf(sy_v_summary____,type,
    summary2: vEBT_VEBT ).

thf(sy_v_treeList_H____,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_treeList____,type,
    treeList2: list_VEBT_VEBT ).

% Relevant facts (10206)
thf(fact_0__092_060open_062summary_A_061_Asummary_H_092_060close_062,axiom,
    summary2 = summary ).

% \<open>summary = summary'\<close>
thf(fact_1_case4_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% case4(9)
thf(fact_2_infsplit,axiom,
    ( info
    = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) ) ).

% infsplit
thf(fact_3_case4_I13_J,axiom,
    ( ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) )
    = ( vEBT_VEBT_set_vebt @ sa ) ) ).

% case4(13)
thf(fact_4_case4_I12_J,axiom,
    vEBT_invar_vebt @ sa @ deg ).

% case4(12)
thf(fact_5_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_6_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_7_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_8_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_9__092_060open_062treeList_A_061_AtreeList_H_092_060close_062,axiom,
    treeList2 = treeList ).

% \<open>treeList = treeList'\<close>
thf(fact_10_prod_Oinject,axiom,
    ! [X1: set_Pr1261947904930325089at_nat,X2: set_Pr1261947904930325089at_nat,Y1: set_Pr1261947904930325089at_nat,Y2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ X1 @ X2 )
        = ( produc2922128104949294807at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: num,X2: num,Y1: num,Y2: num] :
      ( ( ( product_Pair_num_num @ X1 @ X2 )
        = ( product_Pair_num_num @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: nat,X2: num,Y1: nat,Y2: num] :
      ( ( ( product_Pair_nat_num @ X1 @ X2 )
        = ( product_Pair_nat_num @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: nat,X2: nat,Y1: nat,Y2: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X2 )
        = ( product_Pair_nat_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_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_15_old_Oprod_Oinject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList2 ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ).

% case4(8)
thf(fact_21__092_060open_062set__vebt_H_Asummary_A_061_Aset__vebt_H_Asummary_H_092_060close_062,axiom,
    ( ( vEBT_VEBT_set_vebt @ summary2 )
    = ( vEBT_VEBT_set_vebt @ summary ) ) ).

% \<open>set_vebt' summary = set_vebt' summary'\<close>
thf(fact_22__092_060open_062_092_060And_062x_O_Avebt__member_Asummary_Ax_A_061_Avebt__member_Asummary_H_Ax_092_060close_062,axiom,
    ! [X3: nat] :
      ( ( vEBT_vebt_member @ summary2 @ X3 )
      = ( vEBT_vebt_member @ summary @ X3 ) ) ).

% \<open>\<And>x. vebt_member summary x = vebt_member summary' x\<close>
thf(fact_23_aa,axiom,
    ord_less_eq_set_nat @ ( insert_nat @ mi @ ( insert_nat @ ma @ bot_bot_set_nat ) ) @ ( vEBT_VEBT_set_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).

% aa
thf(fact_24_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3843707927480180839at_nat] :
      ~ ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( Y
         != ( produc2922128104949294807at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_26_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_num_num] :
      ~ ! [A3: num,B3: num] :
          ( Y
         != ( product_Pair_num_num @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_27_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_num] :
      ~ ! [A3: nat,B3: num] :
          ( Y
         != ( product_Pair_nat_num @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_28_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_29_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_int_int] :
      ~ ! [A3: int,B3: int] :
          ( Y
         != ( product_Pair_int_int @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_30_deg__deg__n,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_31_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_32_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_33_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs )
             => ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ) ).

% max_in_set_def
thf(fact_34_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y3: nat] :
              ( ( member_nat @ Y3 @ Xs )
             => ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% min_in_set_def
thf(fact_35_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X3 ) @ N ) ) ).

% insert'_pres_valid
thf(fact_36_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_37_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% lesseq_shift
thf(fact_38_case4_I3_J,axiom,
    vEBT_invar_vebt @ summary2 @ m ).

% case4(3)
thf(fact_39_case4_I1_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ( ( vEBT_invar_vebt @ X @ na )
        & ! [Xa: vEBT_VEBT] :
            ( ( vEBT_invar_vebt @ Xa @ na )
           => ( ( ( vEBT_VEBT_set_vebt @ X )
                = ( vEBT_VEBT_set_vebt @ Xa ) )
             => ( Xa = X ) ) ) ) ) ).

% case4(1)
thf(fact_40_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_41_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_42_Pair__inject,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( produc2922128104949294807at_nat @ A @ B )
        = ( produc2922128104949294807at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_43_Pair__inject,axiom,
    ! [A: num,B: num,A2: num,B2: num] :
      ( ( ( product_Pair_num_num @ A @ B )
        = ( product_Pair_num_num @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_44_Pair__inject,axiom,
    ! [A: nat,B: num,A2: nat,B2: num] :
      ( ( ( product_Pair_nat_num @ A @ B )
        = ( product_Pair_nat_num @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_45_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_46_Pair__inject,axiom,
    ! [A: int,B: int,A2: int,B2: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_47_prod__cases,axiom,
    ! [P: produc3843707927480180839at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P @ ( produc2922128104949294807at_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_48_prod__cases,axiom,
    ! [P: product_prod_num_num > $o,P2: product_prod_num_num] :
      ( ! [A3: num,B3: num] : ( P @ ( product_Pair_num_num @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_49_prod__cases,axiom,
    ! [P: product_prod_nat_num > $o,P2: product_prod_nat_num] :
      ( ! [A3: nat,B3: num] : ( P @ ( product_Pair_nat_num @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_50_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_51_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_52_surj__pair,axiom,
    ! [P2: produc3843707927480180839at_nat] :
    ? [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
      ( P2
      = ( produc2922128104949294807at_nat @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_53_surj__pair,axiom,
    ! [P2: product_prod_num_num] :
    ? [X5: num,Y4: num] :
      ( P2
      = ( product_Pair_num_num @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_54_surj__pair,axiom,
    ! [P2: product_prod_nat_num] :
    ? [X5: nat,Y4: num] :
      ( P2
      = ( product_Pair_nat_num @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_55_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X5: nat,Y4: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_56_surj__pair,axiom,
    ! [P2: product_prod_int_int] :
    ? [X5: int,Y4: int] :
      ( P2
      = ( product_Pair_int_int @ X5 @ Y4 ) ) ).

% surj_pair
thf(fact_57_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_58_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_59_singleton__insert__inj__eq,axiom,
    ! [B: $o,A: $o,A4: set_o] :
      ( ( ( insert_o @ B @ bot_bot_set_o )
        = ( insert_o @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_60_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A4: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_61_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A4: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A4 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_62_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ A4 )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_63_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A4 )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_64_singleton__insert__inj__eq_H,axiom,
    ! [A: $o,A4: set_o,B: $o] :
      ( ( ( insert_o @ A @ A4 )
        = ( insert_o @ B @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ bot_bot_set_o ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_65_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A4: set_int,B: int] :
      ( ( ( insert_int @ A @ A4 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_66_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A4: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A4 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_67_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_68_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( member_o @ A @ ( collect_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_74_mem__Collect__eq,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_75_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_76_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A4: set_o] :
      ( ( collect_o
        @ ^ [X4: $o] : ( member_o @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_79_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_80_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( member_complex @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_84_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_85_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_86_Collect__cong,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collec3392354462482085612at_nat @ P )
        = ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_87_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_88_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_89_insert__subset,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) @ B4 )
      = ( ( member8440522571783428010at_nat @ X3 @ B4 )
        & ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_90_insert__subset,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) @ B4 )
      = ( ( member8757157785044589968at_nat @ X3 @ B4 )
        & ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_91_insert__subset,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( insert_complex @ X3 @ A4 ) @ B4 )
      = ( ( member_complex @ X3 @ B4 )
        & ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_92_insert__subset,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
      = ( ( member_real @ X3 @ B4 )
        & ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_93_insert__subset,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
      = ( ( member_o @ X3 @ B4 )
        & ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_94_insert__subset,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A4 ) @ B4 )
      = ( ( member_int @ X3 @ B4 )
        & ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_95_insert__subset,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
      = ( ( member_nat @ X3 @ B4 )
        & ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% insert_subset
thf(fact_96_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_97_singletonI,axiom,
    ! [A: complex] : ( member_complex @ A @ ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singletonI
thf(fact_98_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_99_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_100_singletonI,axiom,
    ! [A: $o] : ( member_o @ A @ ( insert_o @ A @ bot_bot_set_o ) ) ).

% singletonI
thf(fact_101_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_102_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_103_subset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_104_subset__empty,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ bot_bot_set_o )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_empty
thf(fact_105_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_106_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_107_empty__subsetI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A4 ) ).

% empty_subsetI
thf(fact_108_empty__subsetI,axiom,
    ! [A4: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A4 ) ).

% empty_subsetI
thf(fact_109_empty__subsetI,axiom,
    ! [A4: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A4 ) ).

% empty_subsetI
thf(fact_110_empty__subsetI,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A4 ) ).

% empty_subsetI
thf(fact_111_case4_I2_J,axiom,
    ! [S: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ S @ m )
     => ( ( ( vEBT_VEBT_set_vebt @ summary2 )
          = ( vEBT_VEBT_set_vebt @ S ) )
       => ( S = summary2 ) ) ) ).

% case4(2)
thf(fact_112_aca,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( size_s6755466524823107622T_VEBT @ treeList2 ) ) ).

% aca
thf(fact_113_subset__singletonD,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
     => ( ( A4 = bot_bo228742789529271731at_nat )
        | ( A4
          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_114_subset__singletonD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
     => ( ( A4 = bot_bo2099793752762293965at_nat )
        | ( A4
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_115_subset__singletonD,axiom,
    ! [A4: set_o,X3: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) )
     => ( ( A4 = bot_bot_set_o )
        | ( A4
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% subset_singletonD
thf(fact_116_subset__singletonD,axiom,
    ! [A4: set_int,X3: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) )
     => ( ( A4 = bot_bot_set_int )
        | ( A4
          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_117_subset__singletonD,axiom,
    ! [A4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
     => ( ( A4 = bot_bot_set_nat )
        | ( A4
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_118_subset__singleton__iff,axiom,
    ! [X6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X6 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( ( X6 = bot_bo228742789529271731at_nat )
        | ( X6
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_119_subset__singleton__iff,axiom,
    ! [X6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X6 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X6 = bot_bo2099793752762293965at_nat )
        | ( X6
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_120_subset__singleton__iff,axiom,
    ! [X6: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ X6 @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( ( X6 = bot_bot_set_o )
        | ( X6
          = ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% subset_singleton_iff
thf(fact_121_subset__singleton__iff,axiom,
    ! [X6: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X6 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X6 = bot_bot_set_int )
        | ( X6
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_122_subset__singleton__iff,axiom,
    ! [X6: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X6 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X6 = bot_bot_set_nat )
        | ( X6
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_123_empty__Collect__eq,axiom,
    ! [P: complex > $o] :
      ( ( bot_bot_set_complex
        = ( collect_complex @ P ) )
      = ( ! [X4: complex] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_124_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_125_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_126_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_127_empty__Collect__eq,axiom,
    ! [P: $o > $o] :
      ( ( bot_bot_set_o
        = ( collect_o @ P ) )
      = ( ! [X4: $o] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_128_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_129_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% empty_Collect_eq
thf(fact_130_Collect__empty__eq,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( ! [X4: complex] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_131_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X4: set_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_132_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X4: list_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_133_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_134_Collect__empty__eq,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_135_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_136_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ~ ( P @ X4 ) ) ) ).

% Collect_empty_eq
thf(fact_137_all__not__in__conv,axiom,
    ! [A4: set_complex] :
      ( ( ! [X4: complex] :
            ~ ( member_complex @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_complex ) ) ).

% all_not_in_conv
thf(fact_138_all__not__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ! [X4: real] :
            ~ ( member_real @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_139_all__not__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ! [X4: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_140_all__not__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ! [X4: $o] :
            ~ ( member_o @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% all_not_in_conv
thf(fact_141_all__not__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ! [X4: nat] :
            ~ ( member_nat @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_142_all__not__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ! [X4: int] :
            ~ ( member_int @ X4 @ A4 ) )
      = ( A4 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_143_empty__iff,axiom,
    ! [C: complex] :
      ~ ( member_complex @ C @ bot_bot_set_complex ) ).

% empty_iff
thf(fact_144_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_145_empty__iff,axiom,
    ! [C: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_146_empty__iff,axiom,
    ! [C: $o] :
      ~ ( member_o @ C @ bot_bot_set_o ) ).

% empty_iff
thf(fact_147_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_148_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_149_subset__antisym,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( A4 = B4 ) ) ) ).

% subset_antisym
thf(fact_150_subsetI,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ( member_complex @ X5 @ B4 ) )
     => ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ).

% subsetI
thf(fact_151_subsetI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ( member_real @ X5 @ B4 ) )
     => ( ord_less_eq_set_real @ A4 @ B4 ) ) ).

% subsetI
thf(fact_152_subsetI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( member_o @ X5 @ B4 ) )
     => ( ord_less_eq_set_o @ A4 @ B4 ) ) ).

% subsetI
thf(fact_153_subsetI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ( member_int @ X5 @ B4 ) )
     => ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% subsetI
thf(fact_154_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ( member_nat @ X5 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_155_insert__absorb2,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A4 ) )
      = ( insert_nat @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_156_insert__absorb2,axiom,
    ! [X3: int,A4: set_int] :
      ( ( insert_int @ X3 @ ( insert_int @ X3 @ A4 ) )
      = ( insert_int @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_157_insert__absorb2,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ X3 @ A4 ) )
      = ( insert_o @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_158_insert__absorb2,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ X3 @ A4 ) )
      = ( insert8211810215607154385at_nat @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_159_insert__absorb2,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ X3 @ A4 ) )
      = ( insert9069300056098147895at_nat @ X3 @ A4 ) ) ).

% insert_absorb2
thf(fact_160_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_161_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_162_insert__iff,axiom,
    ! [A: complex,B: complex,A4: set_complex] :
      ( ( member_complex @ A @ ( insert_complex @ B @ A4 ) )
      = ( ( A = B )
        | ( member_complex @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_163_insert__iff,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
      = ( ( A = B )
        | ( member_real @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_164_insert__iff,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
      = ( ( A = B )
        | ( member_o @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_165_insert__iff,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_166_insert__iff,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
      = ( ( A = B )
        | ( member_int @ A @ A4 ) ) ) ).

% insert_iff
thf(fact_167_insertCI,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B4 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_168_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ A @ B4 )
       => ( A = B ) )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_169_insertCI,axiom,
    ! [A: complex,B4: set_complex,B: complex] :
      ( ( ~ ( member_complex @ A @ B4 )
       => ( A = B ) )
     => ( member_complex @ A @ ( insert_complex @ B @ B4 ) ) ) ).

% insertCI
thf(fact_170_insertCI,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B4 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertCI
thf(fact_171_insertCI,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( ~ ( member_o @ A @ B4 )
       => ( A = B ) )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertCI
thf(fact_172_insertCI,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B4 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertCI
thf(fact_173_insertCI,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B4 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B4 ) ) ) ).

% insertCI
thf(fact_174_case4_I5_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% case4(5)
thf(fact_175_case4_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% case4(6)
thf(fact_176_ex__in__conv,axiom,
    ! [A4: set_complex] :
      ( ( ? [X4: complex] : ( member_complex @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_complex ) ) ).

% ex_in_conv
thf(fact_177_ex__in__conv,axiom,
    ! [A4: set_real] :
      ( ( ? [X4: real] : ( member_real @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_178_ex__in__conv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ? [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A4 ) )
      = ( A4 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_179_ex__in__conv,axiom,
    ! [A4: set_o] :
      ( ( ? [X4: $o] : ( member_o @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_o ) ) ).

% ex_in_conv
thf(fact_180_ex__in__conv,axiom,
    ! [A4: set_nat] :
      ( ( ? [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_181_ex__in__conv,axiom,
    ! [A4: set_int] :
      ( ( ? [X4: int] : ( member_int @ X4 @ A4 ) )
      = ( A4 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_182_equals0I,axiom,
    ! [A4: set_complex] :
      ( ! [Y4: complex] :
          ~ ( member_complex @ Y4 @ A4 )
     => ( A4 = bot_bot_set_complex ) ) ).

% equals0I
thf(fact_183_equals0I,axiom,
    ! [A4: set_real] :
      ( ! [Y4: real] :
          ~ ( member_real @ Y4 @ A4 )
     => ( A4 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_184_equals0I,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ! [Y4: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y4 @ A4 )
     => ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_185_equals0I,axiom,
    ! [A4: set_o] :
      ( ! [Y4: $o] :
          ~ ( member_o @ Y4 @ A4 )
     => ( A4 = bot_bot_set_o ) ) ).

% equals0I
thf(fact_186_equals0I,axiom,
    ! [A4: set_nat] :
      ( ! [Y4: nat] :
          ~ ( member_nat @ Y4 @ A4 )
     => ( A4 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_187_equals0I,axiom,
    ! [A4: set_int] :
      ( ! [Y4: int] :
          ~ ( member_int @ Y4 @ A4 )
     => ( A4 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_188_equals0D,axiom,
    ! [A4: set_complex,A: complex] :
      ( ( A4 = bot_bot_set_complex )
     => ~ ( member_complex @ A @ A4 ) ) ).

% equals0D
thf(fact_189_equals0D,axiom,
    ! [A4: set_real,A: real] :
      ( ( A4 = bot_bot_set_real )
     => ~ ( member_real @ A @ A4 ) ) ).

% equals0D
thf(fact_190_equals0D,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A4 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_191_equals0D,axiom,
    ! [A4: set_o,A: $o] :
      ( ( A4 = bot_bot_set_o )
     => ~ ( member_o @ A @ A4 ) ) ).

% equals0D
thf(fact_192_equals0D,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( A4 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A4 ) ) ).

% equals0D
thf(fact_193_equals0D,axiom,
    ! [A4: set_int,A: int] :
      ( ( A4 = bot_bot_set_int )
     => ~ ( member_int @ A @ A4 ) ) ).

% equals0D
thf(fact_194_emptyE,axiom,
    ! [A: complex] :
      ~ ( member_complex @ A @ bot_bot_set_complex ) ).

% emptyE
thf(fact_195_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_196_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_197_emptyE,axiom,
    ! [A: $o] :
      ~ ( member_o @ A @ bot_bot_set_o ) ).

% emptyE
thf(fact_198_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_199_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_200_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X4: complex] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_201_Collect__mono__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_202_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X4: set_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_203_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X4: list_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_204_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_205_set__eq__subset,axiom,
    ( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% set_eq_subset
thf(fact_206_subset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ord_less_eq_set_nat @ A4 @ C2 ) ) ) ).

% subset_trans
thf(fact_207_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_208_Collect__mono,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_209_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X5: set_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_210_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_211_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_212_subset__refl,axiom,
    ! [A4: set_nat] : ( ord_less_eq_set_nat @ A4 @ A4 ) ).

% subset_refl
thf(fact_213_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
        ! [T2: complex] :
          ( ( member_complex @ T2 @ A5 )
         => ( member_complex @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_214_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [T2: real] :
          ( ( member_real @ T2 @ A5 )
         => ( member_real @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_215_subset__iff,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A5: set_o,B5: set_o] :
        ! [T2: $o] :
          ( ( member_o @ T2 @ A5 )
         => ( member_o @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_216_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [T2: int] :
          ( ( member_int @ T2 @ A5 )
         => ( member_int @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_217_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat @ T2 @ A5 )
         => ( member_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_218_equalityD2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_nat @ B4 @ A4 ) ) ).

% equalityD2
thf(fact_219_equalityD1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% equalityD1
thf(fact_220_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
        ! [X4: complex] :
          ( ( member_complex @ X4 @ A5 )
         => ( member_complex @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_221_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [X4: real] :
          ( ( member_real @ X4 @ A5 )
         => ( member_real @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_222_subset__eq,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A5: set_o,B5: set_o] :
        ! [X4: $o] :
          ( ( member_o @ X4 @ A5 )
         => ( member_o @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_223_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [X4: int] :
          ( ( member_int @ X4 @ A5 )
         => ( member_int @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_224_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A5 )
         => ( member_nat @ X4 @ B5 ) ) ) ) ).

% subset_eq
thf(fact_225_equalityE,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( A4 = B4 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B4 )
         => ~ ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ).

% equalityE
thf(fact_226_subsetD,axiom,
    ! [A4: set_complex,B4: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B4 )
     => ( ( member_complex @ C @ A4 )
       => ( member_complex @ C @ B4 ) ) ) ).

% subsetD
thf(fact_227_subsetD,axiom,
    ! [A4: set_real,B4: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ( member_real @ C @ A4 )
       => ( member_real @ C @ B4 ) ) ) ).

% subsetD
thf(fact_228_subsetD,axiom,
    ! [A4: set_o,B4: set_o,C: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ( member_o @ C @ A4 )
       => ( member_o @ C @ B4 ) ) ) ).

% subsetD
thf(fact_229_subsetD,axiom,
    ! [A4: set_int,B4: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( member_int @ C @ A4 )
       => ( member_int @ C @ B4 ) ) ) ).

% subsetD
thf(fact_230_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% subsetD
thf(fact_231_in__mono,axiom,
    ! [A4: set_complex,B4: set_complex,X3: complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( member_complex @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_232_in__mono,axiom,
    ! [A4: set_real,B4: set_real,X3: real] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ( member_real @ X3 @ A4 )
       => ( member_real @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_233_in__mono,axiom,
    ! [A4: set_o,B4: set_o,X3: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ( member_o @ X3 @ A4 )
       => ( member_o @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_234_in__mono,axiom,
    ! [A4: set_int,B4: set_int,X3: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ( member_int @ X3 @ A4 )
       => ( member_int @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_235_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( member_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_236_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ? [B6: set_Pr1261947904930325089at_nat] :
          ( ( A4
            = ( insert8211810215607154385at_nat @ A @ B6 ) )
          & ~ ( member8440522571783428010at_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_237_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ? [B6: set_Pr4329608150637261639at_nat] :
          ( ( A4
            = ( insert9069300056098147895at_nat @ A @ B6 ) )
          & ~ ( member8757157785044589968at_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_238_mk__disjoint__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( ( member_complex @ A @ A4 )
     => ? [B6: set_complex] :
          ( ( A4
            = ( insert_complex @ A @ B6 ) )
          & ~ ( member_complex @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_239_mk__disjoint__insert,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ? [B6: set_real] :
          ( ( A4
            = ( insert_real @ A @ B6 ) )
          & ~ ( member_real @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_240_mk__disjoint__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ? [B6: set_o] :
          ( ( A4
            = ( insert_o @ A @ B6 ) )
          & ~ ( member_o @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_241_mk__disjoint__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ? [B6: set_nat] :
          ( ( A4
            = ( insert_nat @ A @ B6 ) )
          & ~ ( member_nat @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_242_mk__disjoint__insert,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ? [B6: set_int] :
          ( ( A4
            = ( insert_int @ A @ B6 ) )
          & ~ ( member_int @ A @ B6 ) ) ) ).

% mk_disjoint_insert
thf(fact_243_insert__commute,axiom,
    ! [X3: nat,Y: nat,A4: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y @ A4 ) )
      = ( insert_nat @ Y @ ( insert_nat @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_244_insert__commute,axiom,
    ! [X3: int,Y: int,A4: set_int] :
      ( ( insert_int @ X3 @ ( insert_int @ Y @ A4 ) )
      = ( insert_int @ Y @ ( insert_int @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_245_insert__commute,axiom,
    ! [X3: $o,Y: $o,A4: set_o] :
      ( ( insert_o @ X3 @ ( insert_o @ Y @ A4 ) )
      = ( insert_o @ Y @ ( insert_o @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_246_insert__commute,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y @ A4 ) )
      = ( insert8211810215607154385at_nat @ Y @ ( insert8211810215607154385at_nat @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_247_insert__commute,axiom,
    ! [X3: produc3843707927480180839at_nat,Y: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y @ A4 ) )
      = ( insert9069300056098147895at_nat @ Y @ ( insert9069300056098147895at_nat @ X3 @ A4 ) ) ) ).

% insert_commute
thf(fact_248_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ B @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ A @ A4 )
            = ( insert8211810215607154385at_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_Pr1261947904930325089at_nat] :
                  ( ( A4
                    = ( insert8211810215607154385at_nat @ B @ C3 ) )
                  & ~ ( member8440522571783428010at_nat @ B @ C3 )
                  & ( B4
                    = ( insert8211810215607154385at_nat @ A @ C3 ) )
                  & ~ ( member8440522571783428010at_nat @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_249_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ B @ B4 )
       => ( ( ( insert9069300056098147895at_nat @ A @ A4 )
            = ( insert9069300056098147895at_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_Pr4329608150637261639at_nat] :
                  ( ( A4
                    = ( insert9069300056098147895at_nat @ B @ C3 ) )
                  & ~ ( member8757157785044589968at_nat @ B @ C3 )
                  & ( B4
                    = ( insert9069300056098147895at_nat @ A @ C3 ) )
                  & ~ ( member8757157785044589968at_nat @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_250_insert__eq__iff,axiom,
    ! [A: complex,A4: set_complex,B: complex,B4: set_complex] :
      ( ~ ( member_complex @ A @ A4 )
     => ( ~ ( member_complex @ B @ B4 )
       => ( ( ( insert_complex @ A @ A4 )
            = ( insert_complex @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_complex] :
                  ( ( A4
                    = ( insert_complex @ B @ C3 ) )
                  & ~ ( member_complex @ B @ C3 )
                  & ( B4
                    = ( insert_complex @ A @ C3 ) )
                  & ~ ( member_complex @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_251_insert__eq__iff,axiom,
    ! [A: real,A4: set_real,B: real,B4: set_real] :
      ( ~ ( member_real @ A @ A4 )
     => ( ~ ( member_real @ B @ B4 )
       => ( ( ( insert_real @ A @ A4 )
            = ( insert_real @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_real] :
                  ( ( A4
                    = ( insert_real @ B @ C3 ) )
                  & ~ ( member_real @ B @ C3 )
                  & ( B4
                    = ( insert_real @ A @ C3 ) )
                  & ~ ( member_real @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_252_insert__eq__iff,axiom,
    ! [A: $o,A4: set_o,B: $o,B4: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ~ ( member_o @ B @ B4 )
       => ( ( ( insert_o @ A @ A4 )
            = ( insert_o @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A = ~ B )
             => ? [C3: set_o] :
                  ( ( A4
                    = ( insert_o @ B @ C3 ) )
                  & ~ ( member_o @ B @ C3 )
                  & ( B4
                    = ( insert_o @ A @ C3 ) )
                  & ~ ( member_o @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_253_insert__eq__iff,axiom,
    ! [A: nat,A4: set_nat,B: nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ~ ( member_nat @ B @ B4 )
       => ( ( ( insert_nat @ A @ A4 )
            = ( insert_nat @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_nat] :
                  ( ( A4
                    = ( insert_nat @ B @ C3 ) )
                  & ~ ( member_nat @ B @ C3 )
                  & ( B4
                    = ( insert_nat @ A @ C3 ) )
                  & ~ ( member_nat @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_254_insert__eq__iff,axiom,
    ! [A: int,A4: set_int,B: int,B4: set_int] :
      ( ~ ( member_int @ A @ A4 )
     => ( ~ ( member_int @ B @ B4 )
       => ( ( ( insert_int @ A @ A4 )
            = ( insert_int @ B @ B4 ) )
          = ( ( ( A = B )
             => ( A4 = B4 ) )
            & ( ( A != B )
             => ? [C3: set_int] :
                  ( ( A4
                    = ( insert_int @ B @ C3 ) )
                  & ~ ( member_int @ B @ C3 )
                  & ( B4
                    = ( insert_int @ A @ C3 ) )
                  & ~ ( member_int @ A @ C3 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_255_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( insert8211810215607154385at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_256_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( insert9069300056098147895at_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_257_insert__absorb,axiom,
    ! [A: complex,A4: set_complex] :
      ( ( member_complex @ A @ A4 )
     => ( ( insert_complex @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_258_insert__absorb,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_259_insert__absorb,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_260_insert__absorb,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_261_insert__absorb,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ A4 )
        = A4 ) ) ).

% insert_absorb
thf(fact_262_insert__ident,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ~ ( member8440522571783428010at_nat @ X3 @ B4 )
       => ( ( ( insert8211810215607154385at_nat @ X3 @ A4 )
            = ( insert8211810215607154385at_nat @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_263_insert__ident,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
     => ( ~ ( member8757157785044589968at_nat @ X3 @ B4 )
       => ( ( ( insert9069300056098147895at_nat @ X3 @ A4 )
            = ( insert9069300056098147895at_nat @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_264_insert__ident,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ ( member_complex @ X3 @ A4 )
     => ( ~ ( member_complex @ X3 @ B4 )
       => ( ( ( insert_complex @ X3 @ A4 )
            = ( insert_complex @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_265_insert__ident,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ~ ( member_real @ X3 @ B4 )
       => ( ( ( insert_real @ X3 @ A4 )
            = ( insert_real @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_266_insert__ident,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ~ ( member_o @ X3 @ B4 )
       => ( ( ( insert_o @ X3 @ A4 )
            = ( insert_o @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_267_insert__ident,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ~ ( member_nat @ X3 @ B4 )
       => ( ( ( insert_nat @ X3 @ A4 )
            = ( insert_nat @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_268_insert__ident,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X3 @ A4 )
     => ( ~ ( member_int @ X3 @ B4 )
       => ( ( ( insert_int @ X3 @ A4 )
            = ( insert_int @ X3 @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% insert_ident
thf(fact_269_Set_Oset__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ A4 )
     => ~ ! [B6: set_Pr1261947904930325089at_nat] :
            ( ( A4
              = ( insert8211810215607154385at_nat @ X3 @ B6 ) )
           => ( member8440522571783428010at_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_270_Set_Oset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ A4 )
     => ~ ! [B6: set_Pr4329608150637261639at_nat] :
            ( ( A4
              = ( insert9069300056098147895at_nat @ X3 @ B6 ) )
           => ( member8757157785044589968at_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_271_Set_Oset__insert,axiom,
    ! [X3: complex,A4: set_complex] :
      ( ( member_complex @ X3 @ A4 )
     => ~ ! [B6: set_complex] :
            ( ( A4
              = ( insert_complex @ X3 @ B6 ) )
           => ( member_complex @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_272_Set_Oset__insert,axiom,
    ! [X3: real,A4: set_real] :
      ( ( member_real @ X3 @ A4 )
     => ~ ! [B6: set_real] :
            ( ( A4
              = ( insert_real @ X3 @ B6 ) )
           => ( member_real @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_273_Set_Oset__insert,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( member_o @ X3 @ A4 )
     => ~ ! [B6: set_o] :
            ( ( A4
              = ( insert_o @ X3 @ B6 ) )
           => ( member_o @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_274_Set_Oset__insert,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( member_nat @ X3 @ A4 )
     => ~ ! [B6: set_nat] :
            ( ( A4
              = ( insert_nat @ X3 @ B6 ) )
           => ( member_nat @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_275_Set_Oset__insert,axiom,
    ! [X3: int,A4: set_int] :
      ( ( member_int @ X3 @ A4 )
     => ~ ! [B6: set_int] :
            ( ( A4
              = ( insert_int @ X3 @ B6 ) )
           => ( member_int @ X3 @ B6 ) ) ) ).

% Set.set_insert
thf(fact_276_insertI2,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ B4 )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_277_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ A @ B4 )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_278_insertI2,axiom,
    ! [A: complex,B4: set_complex,B: complex] :
      ( ( member_complex @ A @ B4 )
     => ( member_complex @ A @ ( insert_complex @ B @ B4 ) ) ) ).

% insertI2
thf(fact_279_insertI2,axiom,
    ! [A: real,B4: set_real,B: real] :
      ( ( member_real @ A @ B4 )
     => ( member_real @ A @ ( insert_real @ B @ B4 ) ) ) ).

% insertI2
thf(fact_280_insertI2,axiom,
    ! [A: $o,B4: set_o,B: $o] :
      ( ( member_o @ A @ B4 )
     => ( member_o @ A @ ( insert_o @ B @ B4 ) ) ) ).

% insertI2
thf(fact_281_insertI2,axiom,
    ! [A: nat,B4: set_nat,B: nat] :
      ( ( member_nat @ A @ B4 )
     => ( member_nat @ A @ ( insert_nat @ B @ B4 ) ) ) ).

% insertI2
thf(fact_282_insertI2,axiom,
    ! [A: int,B4: set_int,B: int] :
      ( ( member_int @ A @ B4 )
     => ( member_int @ A @ ( insert_int @ B @ B4 ) ) ) ).

% insertI2
thf(fact_283_insertI1,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_284_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_285_insertI1,axiom,
    ! [A: complex,B4: set_complex] : ( member_complex @ A @ ( insert_complex @ A @ B4 ) ) ).

% insertI1
thf(fact_286_insertI1,axiom,
    ! [A: real,B4: set_real] : ( member_real @ A @ ( insert_real @ A @ B4 ) ) ).

% insertI1
thf(fact_287_insertI1,axiom,
    ! [A: $o,B4: set_o] : ( member_o @ A @ ( insert_o @ A @ B4 ) ) ).

% insertI1
thf(fact_288_insertI1,axiom,
    ! [A: nat,B4: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B4 ) ) ).

% insertI1
thf(fact_289_insertI1,axiom,
    ! [A: int,B4: set_int] : ( member_int @ A @ ( insert_int @ A @ B4 ) ) ).

% insertI1
thf(fact_290_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_291_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member8757157785044589968at_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_292_insertE,axiom,
    ! [A: complex,B: complex,A4: set_complex] :
      ( ( member_complex @ A @ ( insert_complex @ B @ A4 ) )
     => ( ( A != B )
       => ( member_complex @ A @ A4 ) ) ) ).

% insertE
thf(fact_293_insertE,axiom,
    ! [A: real,B: real,A4: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A4 ) )
     => ( ( A != B )
       => ( member_real @ A @ A4 ) ) ) ).

% insertE
thf(fact_294_insertE,axiom,
    ! [A: $o,B: $o,A4: set_o] :
      ( ( member_o @ A @ ( insert_o @ B @ A4 ) )
     => ( ( A = ~ B )
       => ( member_o @ A @ A4 ) ) ) ).

% insertE
thf(fact_295_insertE,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A4 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A4 ) ) ) ).

% insertE
thf(fact_296_insertE,axiom,
    ! [A: int,B: int,A4: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A4 ) )
     => ( ( A != B )
       => ( member_int @ A @ A4 ) ) ) ).

% insertE
thf(fact_297_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_298_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_299_singleton__inject,axiom,
    ! [A: $o,B: $o] :
      ( ( ( insert_o @ A @ bot_bot_set_o )
        = ( insert_o @ B @ bot_bot_set_o ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_300_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_301_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_302_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ A4 )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_303_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ A4 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_304_insert__not__empty,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ A4 )
     != bot_bot_set_o ) ).

% insert_not_empty
thf(fact_305_insert__not__empty,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ A4 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_306_insert__not__empty,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ A4 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_307_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_308_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_309_doubleton__eq__iff,axiom,
    ! [A: $o,B: $o,C: $o,D: $o] :
      ( ( ( insert_o @ A @ ( insert_o @ B @ bot_bot_set_o ) )
        = ( insert_o @ C @ ( insert_o @ D @ bot_bot_set_o ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_310_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_311_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_312_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_313_singleton__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_314_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_315_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_316_singleton__iff,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_317_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_318_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_319_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_320_singletonD,axiom,
    ! [B: complex,A: complex] :
      ( ( member_complex @ B @ ( insert_complex @ A @ bot_bot_set_complex ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_321_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_322_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_323_singletonD,axiom,
    ! [B: $o,A: $o] :
      ( ( member_o @ B @ ( insert_o @ A @ bot_bot_set_o ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_324_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_325_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_326_subset__insertI2,axiom,
    ! [A4: set_int,B4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_327_subset__insertI2,axiom,
    ! [A4: set_o,B4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_328_subset__insertI2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_329_subset__insertI2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_330_subset__insertI2,axiom,
    ! [A4: set_nat,B4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_331_subset__insertI,axiom,
    ! [B4: set_int,A: int] : ( ord_less_eq_set_int @ B4 @ ( insert_int @ A @ B4 ) ) ).

% subset_insertI
thf(fact_332_subset__insertI,axiom,
    ! [B4: set_o,A: $o] : ( ord_less_eq_set_o @ B4 @ ( insert_o @ A @ B4 ) ) ).

% subset_insertI
thf(fact_333_subset__insertI,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B4 @ ( insert8211810215607154385at_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_334_subset__insertI,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B4 @ ( insert9069300056098147895at_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_335_subset__insertI,axiom,
    ! [B4: set_nat,A: nat] : ( ord_less_eq_set_nat @ B4 @ ( insert_nat @ A @ B4 ) ) ).

% subset_insertI
thf(fact_336_subset__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ B4 ) )
        = ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_337_subset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
     => ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ B4 ) )
        = ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_338_subset__insert,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ ( member_complex @ X3 @ A4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ ( insert_complex @ X3 @ B4 ) )
        = ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_339_subset__insert,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
        = ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_340_subset__insert,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
        = ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_341_subset__insert,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X3 @ A4 )
     => ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X3 @ B4 ) )
        = ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_342_subset__insert,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
        = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ).

% subset_insert
thf(fact_343_insert__mono,axiom,
    ! [C2: set_int,D2: set_int,A: int] :
      ( ( ord_less_eq_set_int @ C2 @ D2 )
     => ( ord_less_eq_set_int @ ( insert_int @ A @ C2 ) @ ( insert_int @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_344_insert__mono,axiom,
    ! [C2: set_o,D2: set_o,A: $o] :
      ( ( ord_less_eq_set_o @ C2 @ D2 )
     => ( ord_less_eq_set_o @ ( insert_o @ A @ C2 ) @ ( insert_o @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_345_insert__mono,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ D2 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C2 ) @ ( insert8211810215607154385at_nat @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_346_insert__mono,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ D2 )
     => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C2 ) @ ( insert9069300056098147895at_nat @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_347_insert__mono,axiom,
    ! [C2: set_nat,D2: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ C2 @ D2 )
     => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C2 ) @ ( insert_nat @ A @ D2 ) ) ) ).

% insert_mono
thf(fact_348_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X3: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X3 ) ).

% buildup_nothing_in_leaf
thf(fact_349_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_350_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_351_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_352_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_353_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_354_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_355_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_356_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_357_the__elem__eq,axiom,
    ! [X3: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_358_the__elem__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_359_the__elem__eq,axiom,
    ! [X3: $o] :
      ( ( the_elem_o @ ( insert_o @ X3 @ bot_bot_set_o ) )
      = X3 ) ).

% the_elem_eq
thf(fact_360_the__elem__eq,axiom,
    ! [X3: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_361_the__elem__eq,axiom,
    ! [X3: int] :
      ( ( the_elem_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
      = X3 ) ).

% the_elem_eq
thf(fact_362_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_363_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_364_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_365_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_366_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_367_even__odd__cases,axiom,
    ! [X3: nat] :
      ( ! [N2: nat] :
          ( X3
         != ( plus_plus_nat @ N2 @ N2 ) )
     => ~ ! [N2: nat] :
            ( X3
           != ( plus_plus_nat @ N2 @ ( suc @ N2 ) ) ) ) ).

% even_odd_cases
thf(fact_368_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_369_maxbmo,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X3 ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X3 ) ) ).

% maxbmo
thf(fact_370_ac,axiom,
    ! [T: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ H )
     => ( ( vEBT_invar_vebt @ K2 @ H )
       => ( ( ( vEBT_VEBT_set_vebt @ T )
            = ( vEBT_VEBT_set_vebt @ K2 ) )
         => ( ( vEBT_vebt_mint @ T )
            = ( vEBT_vebt_mint @ K2 ) ) ) ) ) ).

% ac
thf(fact_371_ad,axiom,
    ! [T: vEBT_VEBT,H: nat,K2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ H )
     => ( ( vEBT_invar_vebt @ K2 @ H )
       => ( ( ( vEBT_VEBT_set_vebt @ T )
            = ( vEBT_VEBT_set_vebt @ K2 ) )
         => ( ( vEBT_vebt_maxt @ T )
            = ( vEBT_vebt_maxt @ K2 ) ) ) ) ) ).

% ad
thf(fact_372_order__refl,axiom,
    ! [X3: set_nat] : ( ord_less_eq_set_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_373_order__refl,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).

% order_refl
thf(fact_374_order__refl,axiom,
    ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).

% order_refl
thf(fact_375_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_376_order__refl,axiom,
    ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).

% order_refl
thf(fact_377_bot__set__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex @ bot_bot_complex_o ) ) ).

% bot_set_def
thf(fact_378_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_379_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_380_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_381_bot__set__def,axiom,
    ( bot_bot_set_o
    = ( collect_o @ bot_bot_o_o ) ) ).

% bot_set_def
thf(fact_382_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_383_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_384_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_385_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_386_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_387_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_388_le__cases3,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ( ord_less_eq_rat @ X3 @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_rat @ Y @ X3 )
         => ~ ( ord_less_eq_rat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_rat @ X3 @ Z2 )
           => ~ ( ord_less_eq_rat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z2 @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z2 )
               => ~ ( ord_less_eq_rat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_rat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_rat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_389_le__cases3,axiom,
    ! [X3: num,Y: num,Z2: num] :
      ( ( ( ord_less_eq_num @ X3 @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_num @ Y @ X3 )
         => ~ ( ord_less_eq_num @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_num @ X3 @ Z2 )
           => ~ ( ord_less_eq_num @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_num @ Z2 @ Y )
             => ~ ( ord_less_eq_num @ Y @ X3 ) )
           => ( ( ( ord_less_eq_num @ Y @ Z2 )
               => ~ ( ord_less_eq_num @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_num @ Z2 @ X3 )
                 => ~ ( ord_less_eq_num @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_390_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_391_le__cases3,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X3 @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y @ X3 )
         => ~ ( ord_less_eq_int @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X3 @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X3 ) )
           => ( ( ( ord_less_eq_int @ Y @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X3 )
                 => ~ ( ord_less_eq_int @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_392_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y3 )
          & ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_393_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
    = ( ^ [X4: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y3 )
          & ( ord_less_eq_rat @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_394_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
    = ( ^ [X4: num,Y3: num] :
          ( ( ord_less_eq_num @ X4 @ Y3 )
          & ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_395_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_396_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    = ( ^ [X4: int,Y3: int] :
          ( ( ord_less_eq_int @ X4 @ Y3 )
          & ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_397_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_398_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_399_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_400_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_401_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_402_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_403_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_404_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_405_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_406_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_407_order__antisym,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_408_order__antisym,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_less_eq_rat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_409_order__antisym,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ord_less_eq_num @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_410_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_411_order__antisym,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_less_eq_int @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_412_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_413_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_414_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_415_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_416_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_417_order__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_418_order__trans,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z2 )
       => ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_419_order__trans,axiom,
    ! [X3: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_420_order__trans,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_421_order__trans,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_422_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_423_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_424_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_425_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_426_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ( ord_less_eq_set_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_427_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_less_eq_rat @ B7 @ A6 )
          & ( ord_less_eq_rat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_428_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
    = ( ^ [A6: num,B7: num] :
          ( ( ord_less_eq_num @ B7 @ A6 )
          & ( ord_less_eq_num @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_429_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_430_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    = ( ^ [A6: int,B7: int] :
          ( ( ord_less_eq_int @ B7 @ A6 )
          & ( ord_less_eq_int @ A6 @ B7 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_431_dual__order_Oantisym,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_432_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_433_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_434_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_435_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_436_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_437_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_438_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_439_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_440_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_441_antisym,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_442_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_443_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_444_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_445_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_446_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z ) )
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_447_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A6 @ B7 )
          & ( ord_less_eq_rat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_448_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z: num] : ( Y5 = Z ) )
    = ( ^ [A6: num,B7: num] :
          ( ( ord_less_eq_num @ A6 @ B7 )
          & ( ord_less_eq_num @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_449_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_450_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    = ( ^ [A6: int,B7: int] :
          ( ( ord_less_eq_int @ A6 @ B7 )
          & ( ord_less_eq_int @ B7 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_451_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_452_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_453_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,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_454_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,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_455_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_456_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_457_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,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_458_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X5: int,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_459_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_460_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_461_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_462_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_463_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_464_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_465_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: num,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_466_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X5: num,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_467_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X5: num,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_468_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X5: num,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_469_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: nat,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_470_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X5: nat,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_471_order__eq__refl,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_472_order__eq__refl,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 = Y )
     => ( ord_less_eq_rat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_473_order__eq__refl,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 = Y )
     => ( ord_less_eq_num @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_474_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_475_order__eq__refl,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 = Y )
     => ( ord_less_eq_int @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_476_linorder__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
      | ( ord_less_eq_rat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_477_linorder__linear,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
      | ( ord_less_eq_num @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_478_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_479_linorder__linear,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
      | ( ord_less_eq_int @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_480_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_481_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_482_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_483_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,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_484_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_485_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_486_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_487_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_488_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,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_489_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,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_490_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,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_491_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,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_492_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,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_493_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,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_494_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,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_495_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,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_496_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,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_497_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,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_498_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,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_499_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,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_500_linorder__le__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X3 @ Y )
     => ( ord_less_eq_rat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_501_linorder__le__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X3 @ Y )
     => ( ord_less_eq_num @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_502_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_503_linorder__le__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X3 @ Y )
     => ( ord_less_eq_int @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_504_order__antisym__conv,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( ord_less_eq_set_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_505_order__antisym__conv,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_506_order__antisym__conv,axiom,
    ! [Y: num,X3: num] :
      ( ( ord_less_eq_num @ Y @ X3 )
     => ( ( ord_less_eq_num @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_507_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_508_order__antisym__conv,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ( ( ord_less_eq_int @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_509_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_510_bot_Oextremum,axiom,
    ! [A: set_o] : ( ord_less_eq_set_o @ bot_bot_set_o @ A ) ).

% bot.extremum
thf(fact_511_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_512_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_513_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_514_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_515_bot_Oextremum__unique,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
      = ( A = bot_bot_set_o ) ) ).

% bot.extremum_unique
thf(fact_516_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_517_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_518_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_519_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_520_bot_Oextremum__uniqueI,axiom,
    ! [A: set_o] :
      ( ( ord_less_eq_set_o @ A @ bot_bot_set_o )
     => ( A = bot_bot_set_o ) ) ).

% bot.extremum_uniqueI
thf(fact_521_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_522_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_523_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_524_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_525_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_526_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_527_nat__add__left__cancel__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_528_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_529_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_530_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_531_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_532_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_533_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_534_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_535_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_536_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_537_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_538_add__shift,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% add_shift
thf(fact_539_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X4: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X4 )
          | ( vEBT_VEBT_membermima @ T2 @ X4 ) ) ) ) ).

% both_member_options_def
thf(fact_540_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_541_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_542_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_543_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_544_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_545_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_546_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_547_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_548_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_549_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_550_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_551_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_552_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_553_not__None__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( X3 != none_P5556105721700978146at_nat )
      = ( ? [Y3: product_prod_nat_nat] :
            ( X3
            = ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_554_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
      = ( ? [Y3: nat] :
            ( X3
            = ( some_nat @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_555_not__None__eq,axiom,
    ! [X3: option_num] :
      ( ( X3 != none_num )
      = ( ? [Y3: num] :
            ( X3
            = ( some_num @ Y3 ) ) ) ) ).

% not_None_eq
thf(fact_556_not__Some__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( ! [Y3: product_prod_nat_nat] :
            ( X3
           != ( some_P7363390416028606310at_nat @ Y3 ) ) )
      = ( X3 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_557_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ( ! [Y3: nat] :
            ( X3
           != ( some_nat @ Y3 ) ) )
      = ( X3 = none_nat ) ) ).

% not_Some_eq
thf(fact_558_not__Some__eq,axiom,
    ! [X3: option_num] :
      ( ( ! [Y3: num] :
            ( X3
           != ( some_num @ Y3 ) ) )
      = ( X3 = none_num ) ) ).

% not_Some_eq
thf(fact_559_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_560_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc5491161045314408544at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > $o,Uv: option4927543243414619207at_nat] :
          ( X3
         != ( produc3994169339658061776at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat] :
            ( X3
           != ( produc3994169339658061776at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V ) @ 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_561_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc2233624965454879586on_nat] :
      ( ! [Uu: nat > nat > $o,Uv: option_nat] :
          ( X3
         != ( produc4035269172776083154on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > $o,V: nat] :
            ( X3
           != ( produc4035269172776083154on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V ) @ 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_562_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc7036089656553540234on_num] :
      ( ! [Uu: num > num > $o,Uv: option_num] :
          ( X3
         != ( produc3576312749637752826on_num @ Uu @ ( produc8585076106096196333on_num @ none_num @ Uv ) ) )
     => ( ! [Uw: num > num > $o,V: num] :
            ( X3
           != ( produc3576312749637752826on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V ) @ 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_563_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc5542196010084753463at_nat] :
      ( ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
          ( X3
         != ( produc2899441246263362727at_nat @ Uu @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
            ( X3
           != ( produc2899441246263362727at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V ) @ 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_564_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc8306885398267862888on_nat] :
      ( ! [Uu: nat > nat > nat,Uv: option_nat] :
          ( X3
         != ( produc8929957630744042906on_nat @ Uu @ ( produc5098337634421038937on_nat @ none_nat @ Uv ) ) )
     => ( ! [Uw: nat > nat > nat,V: nat] :
            ( X3
           != ( produc8929957630744042906on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V ) @ 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_565_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc1193250871479095198on_num] :
      ( ! [Uu: num > num > num,Uv: option_num] :
          ( X3
         != ( produc5778274026573060048on_num @ Uu @ ( produc8585076106096196333on_num @ none_num @ Uv ) ) )
     => ( ! [Uw: num > num > num,V: num] :
            ( X3
           != ( produc5778274026573060048on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V ) @ 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_566_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_567_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_568_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_569_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_570_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_571_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_572_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X22: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_573_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X22: nat] :
            ( Y
           != ( some_nat @ X22 ) ) ) ).

% option.exhaust
thf(fact_574_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X22: num] :
            ( Y
           != ( some_num @ X22 ) ) ) ).

% option.exhaust
thf(fact_575_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_576_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_577_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_578_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X7: option4927543243414619207at_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_579_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X7: option_nat] : ( P3 @ X7 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_580_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X7: option_num] : ( P3 @ X7 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_581_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_582_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_583_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_584_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_585_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_586_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: num] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_587_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_588_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: num,B3: nat] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_589_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: num,B3: num] :
              ( ( X3
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_590_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_591_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_592_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_593_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_594_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_595_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_596_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_597_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_598_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_599_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_600_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_601_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_602_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A6: real,B7: real] : ( plus_plus_real @ B7 @ A6 ) ) ) ).

% add.commute
thf(fact_603_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A6: rat,B7: rat] : ( plus_plus_rat @ B7 @ A6 ) ) ) ).

% add.commute
thf(fact_604_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A6: nat,B7: nat] : ( plus_plus_nat @ B7 @ A6 ) ) ) ).

% add.commute
thf(fact_605_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A6: int,B7: int] : ( plus_plus_int @ B7 @ A6 ) ) ) ).

% add.commute
thf(fact_606_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_607_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_608_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_609_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_610_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_611_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_612_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_613_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_614_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_615_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_616_group__cancel_Oadd2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( plus_plus_real @ A @ B4 )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_617_group__cancel_Oadd2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( plus_plus_rat @ A @ B4 )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_618_group__cancel_Oadd2,axiom,
    ! [B4: nat,K2: nat,B: nat,A: nat] :
      ( ( B4
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B4 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_619_group__cancel_Oadd2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( plus_plus_int @ A @ B4 )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_620_group__cancel_Oadd1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( plus_plus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_621_group__cancel_Oadd1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( plus_plus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_622_group__cancel_Oadd1,axiom,
    ! [A4: nat,K2: nat,A: nat,B: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( plus_plus_nat @ A4 @ B )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_623_group__cancel_Oadd1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( plus_plus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_624_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_real @ I @ K2 )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_625_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_rat @ I @ K2 )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_626_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I @ K2 )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_627_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( K2 = L ) )
     => ( ( plus_plus_int @ I @ K2 )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_628_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_629_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_630_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_631_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_632_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_633_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y ) )
     => ( X3 = Y ) ) ).

% Suc_inject
thf(fact_634_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_635_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_636_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_637_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_638_le__trans,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K2 )
       => ( ord_less_eq_nat @ I @ K2 ) ) ) ).

% le_trans
thf(fact_639_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_640_size__neq__size__imp__neq,axiom,
    ! [X3: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X3 )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_641_size__neq__size__imp__neq,axiom,
    ! [X3: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X3 )
       != ( size_size_list_o @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_642_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X3 )
       != ( size_size_list_nat @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_643_size__neq__size__imp__neq,axiom,
    ! [X3: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X3 )
       != ( size_size_list_int @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_644_size__neq__size__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( ( ( size_size_num @ X3 )
       != ( size_size_num @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_645_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_646_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_647_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_648_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_649_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_650_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_651_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_652_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_653_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
        ? [C4: nat] :
          ( B7
          = ( plus_plus_nat @ A6 @ C4 ) ) ) ) ).

% le_iff_add
thf(fact_654_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_655_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_656_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_657_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_658_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C5: nat] :
            ( B
           != ( plus_plus_nat @ A @ C5 ) ) ) ).

% less_eqE
thf(fact_659_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_660_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_661_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_662_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_663_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_664_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_665_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_666_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_667_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_668_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_669_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_670_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_671_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_672_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_673_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_674_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_675_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_676_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_677_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_678_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_679_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z3: nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
           => ( R @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_680_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P @ M2 )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ M2 @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_681_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_682_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_683_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_684_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_685_Suc__le__D,axiom,
    ! [N: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
     => ? [M: nat] :
          ( M4
          = ( suc @ M ) ) ) ).

% Suc_le_D
thf(fact_686_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_687_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_688_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_689_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_690_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_691_nat__arith_Osuc1,axiom,
    ! [A4: nat,K2: nat,A: nat] :
      ( ( A4
        = ( plus_plus_nat @ K2 @ A ) )
     => ( ( suc @ A4 )
        = ( plus_plus_nat @ K2 @ ( suc @ A ) ) ) ) ).

% nat_arith.suc1
thf(fact_692_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_693_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_694_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_695_add__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_le_mono1
thf(fact_696_add__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_697_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N2: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N2 ) ) ) ).

% le_Suc_ex
thf(fact_698_add__leD2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ K2 @ N ) ) ).

% add_leD2
thf(fact_699_add__leD1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_700_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_701_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_702_add__leE,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K2 ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% add_leE
thf(fact_703_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_704_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_705_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_706_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_707_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_708_lift__Suc__mono__le,axiom,
    ! [F: nat > set_nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_set_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_709_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_710_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_711_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_712_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ N @ N4 )
       => ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_713_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X3 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_714_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_715_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_716_Collect__empty__eq__bot,axiom,
    ! [P: complex > $o] :
      ( ( ( collect_complex @ P )
        = bot_bot_set_complex )
      = ( P = bot_bot_complex_o ) ) ).

% Collect_empty_eq_bot
thf(fact_717_Collect__empty__eq__bot,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( P = bot_bot_set_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_718_Collect__empty__eq__bot,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( P = bot_bot_list_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_719_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_720_Collect__empty__eq__bot,axiom,
    ! [P: $o > $o] :
      ( ( ( collect_o @ P )
        = bot_bot_set_o )
      = ( P = bot_bot_o_o ) ) ).

% Collect_empty_eq_bot
thf(fact_721_Collect__empty__eq__bot,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( P = bot_bot_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_722_Collect__empty__eq__bot,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( P = bot_bot_int_o ) ) ).

% Collect_empty_eq_bot
thf(fact_723_bot__empty__eq,axiom,
    ( bot_bot_complex_o
    = ( ^ [X4: complex] : ( member_complex @ X4 @ bot_bot_set_complex ) ) ) ).

% bot_empty_eq
thf(fact_724_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X4: real] : ( member_real @ X4 @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_725_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_726_bot__empty__eq,axiom,
    ( bot_bot_o_o
    = ( ^ [X4: $o] : ( member_o @ X4 @ bot_bot_set_o ) ) ) ).

% bot_empty_eq
thf(fact_727_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X4: nat] : ( member_nat @ X4 @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_728_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X4: int] : ( member_int @ X4 @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_729_is__singleton__the__elem,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat] :
          ( A5
          = ( insert9069300056098147895at_nat @ ( the_el221668144340439132at_nat @ A5 ) @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_730_is__singleton__the__elem,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( A5
          = ( insert8211810215607154385at_nat @ ( the_el2281957884133575798at_nat @ A5 ) @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_731_is__singleton__the__elem,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
          ( A5
          = ( insert_o @ ( the_elem_o @ A5 ) @ bot_bot_set_o ) ) ) ) ).

% is_singleton_the_elem
thf(fact_732_is__singleton__the__elem,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
          ( A5
          = ( insert_nat @ ( the_elem_nat @ A5 ) @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_the_elem
thf(fact_733_is__singleton__the__elem,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
          ( A5
          = ( insert_int @ ( the_elem_int @ A5 ) @ bot_bot_set_int ) ) ) ) ).

% is_singleton_the_elem
thf(fact_734_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_735_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_736_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_737_sprop1,axiom,
    ( ( sa
      = ( vEBT_Node @ info @ deg @ treeList @ summary ) )
    & ( deg
      = ( plus_plus_nat @ na @ m ) )
    & ( ( size_s6755466524823107622T_VEBT @ treeList )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
    & ( vEBT_invar_vebt @ summary @ m )
    & ! [X: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ treeList ) )
       => ( vEBT_invar_vebt @ X @ na ) ) ) ).

% sprop1
thf(fact_738_is__singletonI,axiom,
    ! [X3: produc3843707927480180839at_nat] : ( is_sin2937591304547752795at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ).

% is_singletonI
thf(fact_739_is__singletonI,axiom,
    ! [X3: product_prod_nat_nat] : ( is_sin2850979758926227957at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).

% is_singletonI
thf(fact_740_is__singletonI,axiom,
    ! [X3: $o] : ( is_singleton_o @ ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% is_singletonI
thf(fact_741_is__singletonI,axiom,
    ! [X3: nat] : ( is_singleton_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).

% is_singletonI
thf(fact_742_is__singletonI,axiom,
    ! [X3: int] : ( is_singleton_int @ ( insert_int @ X3 @ bot_bot_set_int ) ) ).

% is_singletonI
thf(fact_743_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_12 ) ) ).

% not_min_Null_member
thf(fact_744_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).

% min_Null_member
thf(fact_745_power__shift,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ( power_power_nat @ X3 @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% power_shift
thf(fact_746_case4_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList2 )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% case4(4)
thf(fact_747_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_748_a0,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% a0
thf(fact_749__092_060open_062length_AtreeList_H_A_061_A2_A_094_Am_092_060close_062,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% \<open>length treeList' = 2 ^ m\<close>
thf(fact_750__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062treeList_H_Asummary_H_Ainfo_O_As_A_061_ANode_Ainfo_Adeg_AtreeList_H_Asummary_H_A_092_060and_062_Adeg_A_061_An_A_L_Am_A_092_060and_062_Alength_AtreeList_H_A_061_A2_A_094_Am_A_092_060and_062_Ainvar__vebt_Asummary_H_Am_A_092_060and_062_A_I_092_060forall_062t_092_060in_062set_AtreeList_H_O_Ainvar__vebt_At_An_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Info2: option4927543243414619207at_nat] :
        ~ ( ( sa
            = ( vEBT_Node @ Info2 @ deg @ TreeList3 @ Summary2 ) )
          & ( deg
            = ( plus_plus_nat @ na @ m ) )
          & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
          & ( vEBT_invar_vebt @ Summary2 @ m )
          & ! [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
             => ( vEBT_invar_vebt @ X @ na ) ) ) ).

% \<open>\<And>thesis. (\<And>treeList' summary' info. s = Node info deg treeList' summary' \<and> deg = n + m \<and> length treeList' = 2 ^ m \<and> invar_vebt summary' m \<and> (\<forall>t\<in>set treeList'. invar_vebt t n) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_751_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X3 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Sx ) ) ) ).

% succ_corr
thf(fact_752_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X3 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X3 @ Sx ) ) ) ).

% succ_correct
thf(fact_753_case4_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% case4(10)
thf(fact_754_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_755_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_756_is__singletonI_H,axiom,
    ! [A4: set_complex] :
      ( ( A4 != bot_bot_set_complex )
     => ( ! [X5: complex,Y4: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ( member_complex @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_complex @ A4 ) ) ) ).

% is_singletonI'
thf(fact_757_is__singletonI_H,axiom,
    ! [A4: set_real] :
      ( ( A4 != bot_bot_set_real )
     => ( ! [X5: real,Y4: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ( member_real @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_real @ A4 ) ) ) ).

% is_singletonI'
thf(fact_758_is__singletonI_H,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( A4 != bot_bo2099793752762293965at_nat )
     => ( ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A4 )
           => ( ( member8440522571783428010at_nat @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_sin2850979758926227957at_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_759_is__singletonI_H,axiom,
    ! [A4: set_o] :
      ( ( A4 != bot_bot_set_o )
     => ( ! [X5: $o,Y4: $o] :
            ( ( member_o @ X5 @ A4 )
           => ( ( member_o @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_o @ A4 ) ) ) ).

% is_singletonI'
thf(fact_760_is__singletonI_H,axiom,
    ! [A4: set_nat] :
      ( ( A4 != bot_bot_set_nat )
     => ( ! [X5: nat,Y4: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ( member_nat @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_nat @ A4 ) ) ) ).

% is_singletonI'
thf(fact_761_is__singletonI_H,axiom,
    ! [A4: set_int] :
      ( ( A4 != bot_bot_set_int )
     => ( ! [X5: int,Y4: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ( member_int @ Y4 @ A4 )
             => ( X5 = Y4 ) ) )
       => ( is_singleton_int @ A4 ) ) ) ).

% is_singletonI'
thf(fact_762_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_763_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_764_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_765_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu2 @ none_P5556105721700978146at_nat @ Uv2 )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_766_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: num > num > num,Uv2: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu2 @ none_num @ Uv2 )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_767_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu2: nat > nat > nat,Uv2: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu2 @ none_nat @ Uv2 )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_768_subrelI,axiom,
    ! [R2: set_Pr4329608150637261639at_nat,S: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ R2 )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X5 @ Y4 ) @ S ) )
     => ( ord_le1268244103169919719at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_769_subrelI,axiom,
    ! [R2: set_Pr8218934625190621173um_num,S: set_Pr8218934625190621173um_num] :
      ( ! [X5: num,Y4: num] :
          ( ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ R2 )
         => ( member7279096912039735102um_num @ ( product_Pair_num_num @ X5 @ Y4 ) @ S ) )
     => ( ord_le880128212290418581um_num @ R2 @ S ) ) ).

% subrelI
thf(fact_770_subrelI,axiom,
    ! [R2: set_Pr6200539531224447659at_num,S: set_Pr6200539531224447659at_num] :
      ( ! [X5: nat,Y4: num] :
          ( ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ R2 )
         => ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X5 @ Y4 ) @ S ) )
     => ( ord_le8085105155179020875at_num @ R2 @ S ) ) ).

% subrelI
thf(fact_771_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_772_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X5: int,Y4: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X5 @ Y4 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_773_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_774_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_775_is__singletonE,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( is_sin2937591304547752795at_nat @ A4 )
     => ~ ! [X5: produc3843707927480180839at_nat] :
            ( A4
           != ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) ) ) ).

% is_singletonE
thf(fact_776_is__singletonE,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( is_sin2850979758926227957at_nat @ A4 )
     => ~ ! [X5: product_prod_nat_nat] :
            ( A4
           != ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) ) ) ).

% is_singletonE
thf(fact_777_is__singletonE,axiom,
    ! [A4: set_o] :
      ( ( is_singleton_o @ A4 )
     => ~ ! [X5: $o] :
            ( A4
           != ( insert_o @ X5 @ bot_bot_set_o ) ) ) ).

% is_singletonE
thf(fact_778_is__singletonE,axiom,
    ! [A4: set_nat] :
      ( ( is_singleton_nat @ A4 )
     => ~ ! [X5: nat] :
            ( A4
           != ( insert_nat @ X5 @ bot_bot_set_nat ) ) ) ).

% is_singletonE
thf(fact_779_is__singletonE,axiom,
    ! [A4: set_int] :
      ( ( is_singleton_int @ A4 )
     => ~ ! [X5: int] :
            ( A4
           != ( insert_int @ X5 @ bot_bot_set_int ) ) ) ).

% is_singletonE
thf(fact_780_is__singleton__def,axiom,
    ( is_sin2937591304547752795at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat] :
        ? [X4: produc3843707927480180839at_nat] :
          ( A5
          = ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_781_is__singleton__def,axiom,
    ( is_sin2850979758926227957at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
        ? [X4: product_prod_nat_nat] :
          ( A5
          = ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% is_singleton_def
thf(fact_782_is__singleton__def,axiom,
    ( is_singleton_o
    = ( ^ [A5: set_o] :
        ? [X4: $o] :
          ( A5
          = ( insert_o @ X4 @ bot_bot_set_o ) ) ) ) ).

% is_singleton_def
thf(fact_783_is__singleton__def,axiom,
    ( is_singleton_nat
    = ( ^ [A5: set_nat] :
        ? [X4: nat] :
          ( A5
          = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% is_singleton_def
thf(fact_784_is__singleton__def,axiom,
    ( is_singleton_int
    = ( ^ [A5: set_int] :
        ? [X4: int] :
          ( A5
          = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% is_singleton_def
thf(fact_785_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw2 @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_786_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: num > num > num,V2: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw2 @ ( some_num @ V2 ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_787_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw2: nat > nat > nat,V2: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw2 @ ( some_nat @ V2 ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_788_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [V: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A3: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A3 ) )
               => ! [B3: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B3 ) )
                   => ( Y
                     != ( some_P7363390416028606310at_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_789_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [V: num] :
                ( Xa2
                = ( some_num @ V ) )
           => ( ( Xb = none_num )
             => ( Y != none_num ) ) )
         => ~ ! [A3: num] :
                ( ( Xa2
                  = ( some_num @ A3 ) )
               => ! [B3: num] :
                    ( ( Xb
                      = ( some_num @ B3 ) )
                   => ( Y
                     != ( some_num @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_790_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [V: nat] :
                ( Xa2
                = ( some_nat @ V ) )
           => ( ( Xb = none_nat )
             => ( Y != none_nat ) ) )
         => ~ ! [A3: nat] :
                ( ( Xa2
                  = ( some_nat @ A3 ) )
               => ! [B3: nat] :
                    ( ( Xb
                      = ( some_nat @ B3 ) )
                   => ( Y
                     != ( some_nat @ ( X3 @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_791_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_792_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_793_insert__simp__mima,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ Summary ) @ X3 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_794_succ__min,axiom,
    ! [Deg: nat,X3: nat,Mi: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X3 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_795_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_796_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_797_set__n__deg__not__0,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_798_del__single__cont,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( X3 = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_799_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X3 )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_800_misiz,axiom,
    ! [T: vEBT_VEBT,N: nat,M2: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( some_nat @ M2 )
          = ( vEBT_vebt_mint @ T ) )
       => ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% misiz
thf(fact_801_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_802_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X3 ) @ N ) ) ).

% delete_pres_valid
thf(fact_803_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X3 ) @ Y )
        = ( ( X3 != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_804_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X3 ) @ Y )
        = ( ( X3 != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_805_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M2 )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_806_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_real @ M2 )
        = ( numeral_numeral_real @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_807_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_rat @ M2 )
        = ( numeral_numeral_rat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_808_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_nat @ M2 )
        = ( numeral_numeral_nat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_809_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_int @ M2 )
        = ( numeral_numeral_int @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_810_succ__member,axiom,
    ! [T: vEBT_VEBT,X3: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X3 @ Y )
        & ! [Z4: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z4 )
              & ( ord_less_nat @ X3 @ Z4 ) )
           => ( ord_less_eq_nat @ Y @ Z4 ) ) ) ) ).

% succ_member
thf(fact_811_valid__pres__insert,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_invar_vebt @ ( vEBT_vebt_insert @ T @ X3 ) @ N ) ) ) ).

% valid_pres_insert
thf(fact_812_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_813_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_814_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_815_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_816_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_817_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_818_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_819_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_820_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_821_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_822_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_823_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_824_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_825_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_826_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_827_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_828_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_829_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_830_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_831_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_832_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X3 ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_833_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_834_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X3 ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X3 = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_835_delt__out__of__range,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X3 @ Mi )
        | ( ord_less_nat @ Ma @ X3 ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_836_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_837_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_838_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_839_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_840_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_841_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_842_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_843_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_rat @ N )
        = one_one_rat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_844_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_845_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_846_add__numeral__left,axiom,
    ! [V2: num,W: num,Z2: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_847_add__numeral__left,axiom,
    ! [V2: num,W: num,Z2: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_848_add__numeral__left,axiom,
    ! [V2: num,W: num,Z2: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V2 ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_849_add__numeral__left,axiom,
    ! [V2: num,W: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_850_add__numeral__left,axiom,
    ! [V2: num,W: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_851_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_852_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_853_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_854_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_855_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_865_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_866_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_867_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_868_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_869_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_pred__member,axiom,
    ! [T: vEBT_VEBT,X3: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X3 )
        & ! [Z4: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z4 )
              & ( ord_less_nat @ Z4 @ X3 ) )
           => ( ord_less_eq_nat @ Z4 @ Y ) ) ) ) ).

% pred_member
thf(fact_881_case4_I7_J,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I2 ) @ X8 ) )
        = ( vEBT_V8194947554948674370ptions @ summary2 @ I2 ) ) ) ).

% case4(7)
thf(fact_882_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_883_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_884_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_885_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_886_lt__ex,axiom,
    ! [X3: real] :
    ? [Y4: real] : ( ord_less_real @ Y4 @ X3 ) ).

% lt_ex
thf(fact_887_lt__ex,axiom,
    ! [X3: rat] :
    ? [Y4: rat] : ( ord_less_rat @ Y4 @ X3 ) ).

% lt_ex
thf(fact_888_lt__ex,axiom,
    ! [X3: int] :
    ? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).

% lt_ex
thf(fact_889_gt__ex,axiom,
    ! [X3: real] :
    ? [X_12: real] : ( ord_less_real @ X3 @ X_12 ) ).

% gt_ex
thf(fact_890_gt__ex,axiom,
    ! [X3: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X3 @ X_12 ) ).

% gt_ex
thf(fact_891_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X3 @ X_12 ) ).

% gt_ex
thf(fact_892_gt__ex,axiom,
    ! [X3: int] :
    ? [X_12: int] : ( ord_less_int @ X3 @ X_12 ) ).

% gt_ex
thf(fact_893_dense,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ? [Z3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
          & ( ord_less_real @ Z3 @ Y ) ) ) ).

% dense
thf(fact_894_dense,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
          & ( ord_less_rat @ Z3 @ Y ) ) ) ).

% dense
thf(fact_895_less__imp__neq,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_896_less__imp__neq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_897_less__imp__neq,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_898_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_899_less__imp__neq,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_900_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_901_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_902_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_903_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_904_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_905_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_906_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_907_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_908_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_909_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_910_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_911_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_912_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_913_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_914_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_915_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X5 )
             => ( P @ Y6 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_916_antisym__conv3,axiom,
    ! [Y: real,X3: real] :
      ( ~ ( ord_less_real @ Y @ X3 )
     => ( ( ~ ( ord_less_real @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_917_antisym__conv3,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ ( ord_less_rat @ Y @ X3 )
     => ( ( ~ ( ord_less_rat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_918_antisym__conv3,axiom,
    ! [Y: num,X3: num] :
      ( ~ ( ord_less_num @ Y @ X3 )
     => ( ( ~ ( ord_less_num @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_919_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_920_antisym__conv3,axiom,
    ! [Y: int,X3: int] :
      ( ~ ( ord_less_int @ Y @ X3 )
     => ( ( ~ ( ord_less_int @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_921_linorder__cases,axiom,
    ! [X3: real,Y: real] :
      ( ~ ( ord_less_real @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_real @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_922_linorder__cases,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_rat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_923_linorder__cases,axiom,
    ! [X3: num,Y: num] :
      ( ~ ( ord_less_num @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_num @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_924_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_925_linorder__cases,axiom,
    ! [X3: int,Y: int] :
      ( ~ ( ord_less_int @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_int @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_926_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_927_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_928_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_929_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_930_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_931_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_932_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_933_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_934_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_935_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_936_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X7: nat] : ( P3 @ X7 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N3 )
             => ~ ( P4 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_937_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_938_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_939_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_940_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_941_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_942_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_943_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_944_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_945_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_946_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_947_not__less__iff__gr__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y ) )
      = ( ( ord_less_real @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_948_not__less__iff__gr__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y ) )
      = ( ( ord_less_rat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_949_not__less__iff__gr__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y ) )
      = ( ( ord_less_num @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_950_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ( ord_less_nat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_951_not__less__iff__gr__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y ) )
      = ( ( ord_less_int @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_952_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_953_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_954_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_955_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_956_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_957_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_958_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_959_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_960_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_961_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_962_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_963_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_964_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_965_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_966_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_967_linorder__neqE,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_real @ X3 @ Y )
       => ( ord_less_real @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_968_linorder__neqE,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_rat @ X3 @ Y )
       => ( ord_less_rat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_969_linorder__neqE,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_num @ X3 @ Y )
       => ( ord_less_num @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_970_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_971_linorder__neqE,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_int @ X3 @ Y )
       => ( ord_less_int @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_972_order__less__asym,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ~ ( ord_less_real @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_973_order__less__asym,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ~ ( ord_less_rat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_974_order__less__asym,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ~ ( ord_less_num @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_975_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_976_order__less__asym,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ~ ( ord_less_int @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_977_linorder__neq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( X3 != Y )
      = ( ( ord_less_real @ X3 @ Y )
        | ( ord_less_real @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_978_linorder__neq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( X3 != Y )
      = ( ( ord_less_rat @ X3 @ Y )
        | ( ord_less_rat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_979_linorder__neq__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( X3 != Y )
      = ( ( ord_less_num @ X3 @ Y )
        | ( ord_less_num @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_980_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_981_linorder__neq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( X3 != Y )
      = ( ( ord_less_int @ X3 @ Y )
        | ( ord_less_int @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_982_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_983_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_984_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_985_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_986_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_987_order__less__trans,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_988_order__less__trans,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_989_order__less__trans,axiom,
    ! [X3: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_990_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_991_order__less__trans,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_992_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_993_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_994_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_995_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_996_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_997_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_998_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_999_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1000_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1001_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ 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_1002_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1003_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1004_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1005_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1006_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1007_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1008_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1009_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1010_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1011_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = 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_1012_order__less__irrefl,axiom,
    ! [X3: real] :
      ~ ( ord_less_real @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1013_order__less__irrefl,axiom,
    ! [X3: rat] :
      ~ ( ord_less_rat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1014_order__less__irrefl,axiom,
    ! [X3: num] :
      ~ ( ord_less_num @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1015_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1016_order__less__irrefl,axiom,
    ! [X3: int] :
      ~ ( ord_less_int @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_1017_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1018_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1019_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1020_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1021_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1022_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1023_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1024_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1025_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1026_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,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_1027_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1028_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1029_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1030_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1031_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1032_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1033_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1034_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1035_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1036_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_1037_order__less__not__sym,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ~ ( ord_less_real @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_1038_order__less__not__sym,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ~ ( ord_less_rat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_1039_order__less__not__sym,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ~ ( ord_less_num @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_1040_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_1041_order__less__not__sym,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ~ ( ord_less_int @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_1042_order__less__imp__triv,axiom,
    ! [X3: real,Y: real,P: $o] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ( ord_less_real @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1043_order__less__imp__triv,axiom,
    ! [X3: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ( ord_less_rat @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1044_order__less__imp__triv,axiom,
    ! [X3: num,Y: num,P: $o] :
      ( ( ord_less_num @ X3 @ Y )
     => ( ( ord_less_num @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1045_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1046_order__less__imp__triv,axiom,
    ! [X3: int,Y: int,P: $o] :
      ( ( ord_less_int @ X3 @ Y )
     => ( ( ord_less_int @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1047_linorder__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_real @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_1048_linorder__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_rat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_1049_linorder__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_num @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_1050_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_1051_linorder__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_int @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_1052_order__less__imp__not__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1053_order__less__imp__not__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1054_order__less__imp__not__eq,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1055_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1056_order__less__imp__not__eq,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1057_order__less__imp__not__eq2,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1058_order__less__imp__not__eq2,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1059_order__less__imp__not__eq2,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1060_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1061_order__less__imp__not__eq2,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1062_order__less__imp__not__less,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ~ ( ord_less_real @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1063_order__less__imp__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ~ ( ord_less_rat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1064_order__less__imp__not__less,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ~ ( ord_less_num @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1065_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1066_order__less__imp__not__less,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ~ ( ord_less_int @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1067_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_1068_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_1069_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_1070_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_1071_one__reorient,axiom,
    ! [X3: complex] :
      ( ( one_one_complex = X3 )
      = ( X3 = one_one_complex ) ) ).

% one_reorient
thf(fact_1072_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
      = ( X3 = one_one_real ) ) ).

% one_reorient
thf(fact_1073_one__reorient,axiom,
    ! [X3: rat] :
      ( ( one_one_rat = X3 )
      = ( X3 = one_one_rat ) ) ).

% one_reorient
thf(fact_1074_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
      = ( X3 = one_one_nat ) ) ).

% one_reorient
thf(fact_1075_one__reorient,axiom,
    ! [X3: int] :
      ( ( one_one_int = X3 )
      = ( X3 = one_one_int ) ) ).

% one_reorient
thf(fact_1076_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_1077_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
             => ( P @ M3 ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_1078_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N2 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_1079_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_1080_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_real @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1081_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_rat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1082_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_num @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1083_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1084_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N: nat,N4: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ N @ N4 )
       => ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_1085_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_real @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1086_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_rat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_rat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1087_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_num @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_num @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1088_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1089_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N: nat,M2: nat] :
      ( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_1090_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_1091_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_1092_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_1093_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_1094_order__le__imp__less__or__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ Y )
     => ( ( ord_less_real @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1095_order__le__imp__less__or__eq,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_set_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1096_order__le__imp__less__or__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_less_rat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1097_order__le__imp__less__or__eq,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ord_less_num @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1098_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1099_order__le__imp__less__or__eq,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_less_int @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1100_linorder__le__less__linear,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ Y )
      | ( ord_less_real @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1101_linorder__le__less__linear,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
      | ( ord_less_rat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1102_linorder__le__less__linear,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
      | ( ord_less_num @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1103_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1104_linorder__le__less__linear,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
      | ( ord_less_int @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1105_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1106_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1107_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1108_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1109_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1110_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: real,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_1111_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: rat,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_1112_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: num,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_1113_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: nat,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_1114_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X5: int,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_1115_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1116_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1117_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1118_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1119_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1120_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1121_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1122_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1123_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_1124_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,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_1125_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1126_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1127_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1128_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1129_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1130_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1131_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1132_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_eq_num @ X5 @ Y4 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1133_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: num,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_1134_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: num,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_1135_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y4: real] :
              ( ( ord_less_real @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1136_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y4: rat] :
              ( ( ord_less_rat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1137_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y4: num] :
              ( ( ord_less_num @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1138_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y4: nat] :
              ( ( ord_less_nat @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1139_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y4: int] :
              ( ( ord_less_int @ X5 @ Y4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y4 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1140_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,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_1141_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,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_1142_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,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_1143_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,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_1144_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,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_1145_order__less__le__trans,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1146_order__less__le__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1147_order__less__le__trans,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1148_order__less__le__trans,axiom,
    ! [X3: num,Y: num,Z2: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1149_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1150_order__less__le__trans,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1151_order__le__less__trans,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( ord_less_eq_real @ X3 @ Y )
     => ( ( ord_less_real @ Y @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1152_order__le__less__trans,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_set_nat @ Y @ Z2 )
       => ( ord_less_set_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1153_order__le__less__trans,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_less_rat @ Y @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1154_order__le__less__trans,axiom,
    ! [X3: num,Y: num,Z2: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ord_less_num @ Y @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1155_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1156_order__le__less__trans,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_less_int @ Y @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1157_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1158_order__neq__le__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A != B )
     => ( ( ord_less_eq_set_nat @ A @ B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1159_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1160_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1161_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1162_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1163_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1164_order__le__neq__trans,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1165_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1166_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1167_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1168_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1169_order__less__imp__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ord_less_eq_real @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1170_order__less__imp__le,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X3 @ Y )
     => ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1171_order__less__imp__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ord_less_eq_rat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1172_order__less__imp__le,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_num @ X3 @ Y )
     => ( ord_less_eq_num @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1173_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1174_order__less__imp__le,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ X3 @ Y )
     => ( ord_less_eq_int @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_1175_linorder__not__less,axiom,
    ! [X3: real,Y: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y ) )
      = ( ord_less_eq_real @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_1176_linorder__not__less,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y ) )
      = ( ord_less_eq_rat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_1177_linorder__not__less,axiom,
    ! [X3: num,Y: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y ) )
      = ( ord_less_eq_num @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_1178_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_1179_linorder__not__less,axiom,
    ! [X3: int,Y: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y ) )
      = ( ord_less_eq_int @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_1180_linorder__not__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X3 @ Y ) )
      = ( ord_less_real @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_1181_linorder__not__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X3 @ Y ) )
      = ( ord_less_rat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_1182_linorder__not__le,axiom,
    ! [X3: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X3 @ Y ) )
      = ( ord_less_num @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_1183_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
      = ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_1184_linorder__not__le,axiom,
    ! [X3: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X3 @ Y ) )
      = ( ord_less_int @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_1185_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_eq_real @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1186_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1187_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1188_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y3: num] :
          ( ( ord_less_eq_num @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1189_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1190_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y3: int] :
          ( ( ord_less_eq_int @ X4 @ Y3 )
          & ( X4 != Y3 ) ) ) ) ).

% order_less_le
thf(fact_1191_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_real @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1192_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( ord_less_set_nat @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1193_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y3: rat] :
          ( ( ord_less_rat @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1194_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X4: num,Y3: num] :
          ( ( ord_less_num @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1195_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_nat @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1196_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y3: int] :
          ( ( ord_less_int @ X4 @ Y3 )
          | ( X4 = Y3 ) ) ) ) ).

% order_le_less
thf(fact_1197_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1198_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1199_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1200_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1201_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1202_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1203_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1204_order_Ostrict__implies__order,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1205_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1206_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1207_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1208_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1209_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A6: real] :
          ( ( ord_less_eq_real @ B7 @ A6 )
          & ~ ( ord_less_eq_real @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1210_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ~ ( ord_less_eq_set_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1211_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( ord_less_eq_rat @ B7 @ A6 )
          & ~ ( ord_less_eq_rat @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1212_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A6: num] :
          ( ( ord_less_eq_num @ B7 @ A6 )
          & ~ ( ord_less_eq_num @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1213_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1214_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A6: int] :
          ( ( ord_less_eq_int @ B7 @ A6 )
          & ~ ( ord_less_eq_int @ A6 @ B7 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1215_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1216_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1217_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1218_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1219_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1220_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1221_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1222_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C @ B )
       => ( ord_less_set_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1223_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1224_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1225_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1226_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1227_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B7: real,A6: real] :
          ( ( ord_less_eq_real @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1228_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_eq_set_nat @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1229_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( ord_less_eq_rat @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1230_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B7: num,A6: num] :
          ( ( ord_less_eq_num @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1231_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1232_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B7: int,A6: int] :
          ( ( ord_less_eq_int @ B7 @ A6 )
          & ( A6 != B7 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1233_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B7: real,A6: real] :
          ( ( ord_less_real @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1234_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( ord_less_set_nat @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1235_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( ord_less_rat @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1236_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A6: num] :
          ( ( ord_less_num @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1237_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_less_nat @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1238_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( ( ord_less_int @ B7 @ A6 )
          | ( A6 = B7 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1239_dense__le__bounded,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X3 @ W2 )
           => ( ( ord_less_real @ W2 @ Y )
             => ( ord_less_eq_real @ W2 @ Z2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1240_dense__le__bounded,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X3 @ W2 )
           => ( ( ord_less_rat @ W2 @ Y )
             => ( ord_less_eq_rat @ W2 @ Z2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1241_dense__ge__bounded,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z2 @ W2 )
           => ( ( ord_less_real @ W2 @ X3 )
             => ( ord_less_eq_real @ Y @ W2 ) ) )
       => ( ord_less_eq_real @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1242_dense__ge__bounded,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( ord_less_rat @ Z2 @ X3 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z2 @ W2 )
           => ( ( ord_less_rat @ W2 @ X3 )
             => ( ord_less_eq_rat @ Y @ W2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1243_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A6: real,B7: real] :
          ( ( ord_less_eq_real @ A6 @ B7 )
          & ~ ( ord_less_eq_real @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1244_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_set_nat @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1245_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A6 @ B7 )
          & ~ ( ord_less_eq_rat @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1246_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A6: num,B7: num] :
          ( ( ord_less_eq_num @ A6 @ B7 )
          & ~ ( ord_less_eq_num @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1247_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ~ ( ord_less_eq_nat @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1248_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A6: int,B7: int] :
          ( ( ord_less_eq_int @ A6 @ B7 )
          & ~ ( ord_less_eq_int @ B7 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1249_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1250_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1251_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1252_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1253_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1254_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1255_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1256_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C )
       => ( ord_less_set_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1257_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1258_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1259_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1260_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1261_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A6: real,B7: real] :
          ( ( ord_less_eq_real @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1262_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_eq_set_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1263_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_less_eq_rat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1264_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A6: num,B7: num] :
          ( ( ord_less_eq_num @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1265_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_eq_nat @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1266_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A6: int,B7: int] :
          ( ( ord_less_eq_int @ A6 @ B7 )
          & ( A6 != B7 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1267_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A6: real,B7: real] :
          ( ( ord_less_real @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1268_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( ord_less_set_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1269_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_less_rat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1270_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A6: num,B7: num] :
          ( ( ord_less_num @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1271_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_less_nat @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1272_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] :
          ( ( ord_less_int @ A6 @ B7 )
          | ( A6 = B7 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1273_not__le__imp__less,axiom,
    ! [Y: real,X3: real] :
      ( ~ ( ord_less_eq_real @ Y @ X3 )
     => ( ord_less_real @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_1274_not__le__imp__less,axiom,
    ! [Y: rat,X3: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X3 )
     => ( ord_less_rat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_1275_not__le__imp__less,axiom,
    ! [Y: num,X3: num] :
      ( ~ ( ord_less_eq_num @ Y @ X3 )
     => ( ord_less_num @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_1276_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X3 )
     => ( ord_less_nat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_1277_not__le__imp__less,axiom,
    ! [Y: int,X3: int] :
      ( ~ ( ord_less_eq_int @ Y @ X3 )
     => ( ord_less_int @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_1278_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y3: real] :
          ( ( ord_less_eq_real @ X4 @ Y3 )
          & ~ ( ord_less_eq_real @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1279_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y3 )
          & ~ ( ord_less_eq_set_nat @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1280_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y3: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y3 )
          & ~ ( ord_less_eq_rat @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1281_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y3: num] :
          ( ( ord_less_eq_num @ X4 @ Y3 )
          & ~ ( ord_less_eq_num @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1282_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y3 )
          & ~ ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1283_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y3: int] :
          ( ( ord_less_eq_int @ X4 @ Y3 )
          & ~ ( ord_less_eq_int @ Y3 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1284_dense__le,axiom,
    ! [Y: real,Z2: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y )
         => ( ord_less_eq_real @ X5 @ Z2 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_le
thf(fact_1285_dense__le,axiom,
    ! [Y: rat,Z2: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Y )
         => ( ord_less_eq_rat @ X5 @ Z2 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_le
thf(fact_1286_dense__ge,axiom,
    ! [Z2: real,Y: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z2 @ X5 )
         => ( ord_less_eq_real @ Y @ X5 ) )
     => ( ord_less_eq_real @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_1287_dense__ge,axiom,
    ! [Z2: rat,Y: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ Z2 @ X5 )
         => ( ord_less_eq_rat @ Y @ X5 ) )
     => ( ord_less_eq_rat @ Y @ Z2 ) ) ).

% dense_ge
thf(fact_1288_antisym__conv2,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ Y )
     => ( ( ~ ( ord_less_real @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1289_antisym__conv2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_set_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1290_antisym__conv2,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ~ ( ord_less_rat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1291_antisym__conv2,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ~ ( ord_less_num @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1292_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1293_antisym__conv2,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ~ ( ord_less_int @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_1294_antisym__conv1,axiom,
    ! [X3: real,Y: real] :
      ( ~ ( ord_less_real @ X3 @ Y )
     => ( ( ord_less_eq_real @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1295_antisym__conv1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1296_antisym__conv1,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y )
     => ( ( ord_less_eq_rat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1297_antisym__conv1,axiom,
    ! [X3: num,Y: num] :
      ( ~ ( ord_less_num @ X3 @ Y )
     => ( ( ord_less_eq_num @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1298_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1299_antisym__conv1,axiom,
    ! [X3: int,Y: int] :
      ( ~ ( ord_less_int @ X3 @ Y )
     => ( ( ord_less_eq_int @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_1300_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1301_nless__le,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ~ ( ord_less_set_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1302_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1303_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1304_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1305_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1306_leI,axiom,
    ! [X3: real,Y: real] :
      ( ~ ( ord_less_real @ X3 @ Y )
     => ( ord_less_eq_real @ Y @ X3 ) ) ).

% leI
thf(fact_1307_leI,axiom,
    ! [X3: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y )
     => ( ord_less_eq_rat @ Y @ X3 ) ) ).

% leI
thf(fact_1308_leI,axiom,
    ! [X3: num,Y: num] :
      ( ~ ( ord_less_num @ X3 @ Y )
     => ( ord_less_eq_num @ Y @ X3 ) ) ).

% leI
thf(fact_1309_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% leI
thf(fact_1310_leI,axiom,
    ! [X3: int,Y: int] :
      ( ~ ( ord_less_int @ X3 @ Y )
     => ( ord_less_eq_int @ Y @ X3 ) ) ).

% leI
thf(fact_1311_leD,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ Y @ X3 )
     => ~ ( ord_less_real @ X3 @ Y ) ) ).

% leD
thf(fact_1312_leD,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ~ ( ord_less_set_nat @ X3 @ Y ) ) ).

% leD
thf(fact_1313_leD,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ~ ( ord_less_rat @ X3 @ Y ) ) ).

% leD
thf(fact_1314_leD,axiom,
    ! [Y: num,X3: num] :
      ( ( ord_less_eq_num @ Y @ X3 )
     => ~ ( ord_less_num @ X3 @ Y ) ) ).

% leD
thf(fact_1315_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y ) ) ).

% leD
thf(fact_1316_leD,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ~ ( ord_less_int @ X3 @ Y ) ) ).

% leD
thf(fact_1317_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1318_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1319_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1320_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1321_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1322_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1323_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1324_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1325_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1326_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1327_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1328_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_1329_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1330_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1331_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1332_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1333_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1334_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1335_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1336_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1337_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1338_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1339_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1340_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( K2 = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1341_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( I = J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1342_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1343_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1344_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( I = J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1345_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1346_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1347_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1348_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1349_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1350_bot_Onot__eq__extremum,axiom,
    ! [A: set_o] :
      ( ( A != bot_bot_set_o )
      = ( ord_less_set_o @ bot_bot_set_o @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1351_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_1352_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_1353_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1354_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_1355_bot_Oextremum__strict,axiom,
    ! [A: set_o] :
      ~ ( ord_less_set_o @ A @ bot_bot_set_o ) ).

% bot.extremum_strict
thf(fact_1356_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1357_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1358_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1359_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1360_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_1361_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,K: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K )
                   => ( P @ I3 @ K ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1362_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K2 )
       => ( ord_less_nat @ ( suc @ I ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_1363_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_1364_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_1365_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M6: nat] :
            ( ( M2
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1366_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_1367_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_1368_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_1369_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_1370_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1371_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_1372_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1373_Suc__lessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K2 )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K2
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_1374_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_1375_Nat_OlessE,axiom,
    ! [I: nat,K2: nat] :
      ( ( ord_less_nat @ I @ K2 )
     => ( ( K2
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K2
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1376_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_1377_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1378_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1379_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
          | ( M5 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1380_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_1381_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
          & ( M5 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1382_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_1383_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_1384_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_1385_add__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_less_mono1
thf(fact_1386_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1387_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1388_add__less__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1389_add__lessD1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
     => ( ord_less_nat @ I @ K2 ) ) ).

% add_lessD1
thf(fact_1390_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_1391_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_1392_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_1393_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_1394_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_1395_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_1396_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_1397_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_1398_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_1399_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_1400_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_1401_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_1402_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_1403_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_1404_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1405_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1406_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1407_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1408_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1409_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1410_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1411_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1412_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1413_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I @ J )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1414_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1415_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1416_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I @ J )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1417_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: real,J: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I @ J )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I @ K2 ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1418_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K2 ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1419_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1420_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: int,J: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I @ J )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I @ K2 ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1421_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1422_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1423_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1424_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1425_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1426_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ ( suc @ N2 ) )
                 => ( P @ N2 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_1427_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N2: nat] :
              ( ( ord_less_eq_nat @ I @ N2 )
             => ( ( ord_less_nat @ N2 @ J )
               => ( ( P @ N2 )
                 => ( P @ ( suc @ N2 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_1428_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1429_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1430_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1431_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1432_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_1433_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_1434_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1435_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K2: nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_nat @ M @ N2 )
         => ( ord_less_nat @ ( F @ M ) @ ( F @ N2 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K2 ) @ ( F @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1436_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1437_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1438_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1439_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_1440_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1441_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1442_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1443_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_1444_le__num__One__iff,axiom,
    ! [X3: num] :
      ( ( ord_less_eq_num @ X3 @ one )
      = ( X3 = one ) ) ).

% le_num_One_iff
thf(fact_1445_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_1446_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_1447_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_1448_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_1449_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_1450_Suc__nat__number__of__add,axiom,
    ! [V2: num,N: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ N ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ one ) ) @ N ) ) ).

% Suc_nat_number_of_add
thf(fact_1451_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X3: nat,Mi: nat,TreeList: 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 @ TreeList @ Summary ) @ X3 )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_1452_power__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X3 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_1453_power__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X3 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_1454_power__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X3 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_1455_power__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X3 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_1456_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y3: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% greater_shift
thf(fact_1457_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y3: nat] : ( vEBT_VEBT_less @ ( some_nat @ X4 ) @ ( some_nat @ Y3 ) ) ) ) ).

% less_shift
thf(fact_1458_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X3 )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_1459_power__strict__increasing__iff,axiom,
    ! [B: real,X3: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X3 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1460_power__strict__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X3 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1461_power__strict__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X3 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1462_power__strict__increasing__iff,axiom,
    ! [B: int,X3: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X3 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_1463_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X3: nat,Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X3 )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 )
          = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X3 )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 )
          = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X3 ) ) ) ) ).

% VEBT_internal.insert'.simps(2)
thf(fact_1464_sumtreelistcong,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ X8 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I2 ) ) ) ).

% sumtreelistcong
thf(fact_1465_setcongy,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList2 @ I ) )
        = ( vEBT_VEBT_set_vebt @ ( nth_VEBT_VEBT @ treeList @ I ) ) ) ) ).

% setcongy
thf(fact_1466_membercongy,axiom,
    ! [I: nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList2 @ I ) @ X3 )
        = ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ treeList @ I ) @ X3 ) ) ) ).

% membercongy
thf(fact_1467_ex__power__ivl2,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
       => ? [N2: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_1468_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_1469_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_1470_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_1471_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_1472_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_1473_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_1474_psubsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( A4 != B4 )
       => ( ord_less_set_nat @ A4 @ B4 ) ) ) ).

% psubsetI
thf(fact_1475_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X3 )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 @ Px ) ) ) ).

% pred_corr
thf(fact_1476_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X3 )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X3 @ Sx ) ) ) ).

% pred_correct
thf(fact_1477_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ one_one_rat @ N )
      = one_one_rat ) ).

% power_one
thf(fact_1478_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ one_one_nat @ N )
      = one_one_nat ) ).

% power_one
thf(fact_1479_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_real @ one_one_real @ N )
      = one_one_real ) ).

% power_one
thf(fact_1480_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_int @ one_one_int @ N )
      = one_one_int ) ).

% power_one
thf(fact_1481_power__one,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ one_one_complex @ N )
      = one_one_complex ) ).

% power_one
thf(fact_1482_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1483_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1484_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1485_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_1486_power__inject__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M2 )
          = ( power_power_real @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_1487_power__inject__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M2 )
          = ( power_power_rat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_1488_power__inject__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M2 )
          = ( power_power_nat @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_1489_power__inject__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M2 )
          = ( power_power_int @ A @ N ) )
        = ( M2 = N ) ) ) ).

% power_inject_exp
thf(fact_1490_treecongy,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( nth_VEBT_VEBT @ treeList2 @ I )
        = ( nth_VEBT_VEBT @ treeList @ I ) ) ) ).

% treecongy
thf(fact_1491_not__psubset__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_1492_not__psubset__empty,axiom,
    ! [A4: set_o] :
      ~ ( ord_less_set_o @ A4 @ bot_bot_set_o ) ).

% not_psubset_empty
thf(fact_1493_not__psubset__empty,axiom,
    ! [A4: set_nat] :
      ~ ( ord_less_set_nat @ A4 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1494_not__psubset__empty,axiom,
    ! [A4: set_int] :
      ~ ( ord_less_set_int @ A4 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1495_psubsetE,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ~ ( ( ord_less_eq_set_nat @ A4 @ B4 )
         => ( ord_less_eq_set_nat @ B4 @ A4 ) ) ) ).

% psubsetE
thf(fact_1496_psubset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ( A5 != B5 ) ) ) ) ).

% psubset_eq
thf(fact_1497_psubset__imp__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% psubset_imp_subset
thf(fact_1498_psubset__subset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ord_less_set_nat @ A4 @ C2 ) ) ) ).

% psubset_subset_trans
thf(fact_1499_subset__not__subset__eq,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B5 )
          & ~ ( ord_less_eq_set_nat @ B5 @ A5 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1500_subset__psubset__trans,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_set_nat @ B4 @ C2 )
       => ( ord_less_set_nat @ A4 @ C2 ) ) ) ).

% subset_psubset_trans
thf(fact_1501_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B5 )
          | ( A5 = B5 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1502_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_1503_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_1504_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_1505_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_1506_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_1507_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_1508_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_1509_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_1510_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1511_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1512_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1513_power__strict__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_strict_increasing
thf(fact_1514_power__less__imp__less__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1515_power__less__imp__less__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1516_power__less__imp__less__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1517_power__less__imp__less__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% power_less_imp_less_exp
thf(fact_1518_power__increasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1519_power__increasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1520_power__increasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1521_power__increasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).

% power_increasing
thf(fact_1522_power__le__imp__le__exp,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1523_power__le__imp__le__exp,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1524_power__le__imp__le__exp,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1525_power__le__imp__le__exp,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_1526_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_1527_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_1528_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_1529_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_1530_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_1531_less__exp,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% less_exp
thf(fact_1532_self__le__ge2__pow,axiom,
    ! [K2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ M2 @ ( power_power_nat @ K2 @ M2 ) ) ) ).

% self_le_ge2_pow
thf(fact_1533_power2__nat__le__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_eq_le
thf(fact_1534_power2__nat__le__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% power2_nat_le_imp_le
thf(fact_1535_ex__power__ivl1,axiom,
    ! [B: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K2 )
       => ? [N2: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_1536_semiring__norm_I76_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).

% semiring_norm(76)
thf(fact_1537_semiring__norm_I69_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M2 ) @ one ) ).

% semiring_norm(69)
thf(fact_1538_insert__correct,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 ) )
       => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
          = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X3 ) ) ) ) ) ).

% insert_correct
thf(fact_1539_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_1540_insert__corr,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 ) )
       => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
          = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X3 ) ) ) ) ) ).

% insert_corr
thf(fact_1541_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) @ X3 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_1542_semiring__norm_I75_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_num @ M2 @ one ) ).

% semiring_norm(75)
thf(fact_1543_semiring__norm_I68_J,axiom,
    ! [N: num] : ( ord_less_eq_num @ one @ N ) ).

% semiring_norm(68)
thf(fact_1544_semiring__norm_I78_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(78)
thf(fact_1545_semiring__norm_I71_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(71)
thf(fact_1546_semiring__norm_I87_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( bit0 @ M2 )
        = ( bit0 @ N ) )
      = ( M2 = N ) ) ).

% semiring_norm(87)
thf(fact_1547_UnCI,axiom,
    ! [C: complex,B4: set_complex,A4: set_complex] :
      ( ( ~ ( member_complex @ C @ B4 )
       => ( member_complex @ C @ A4 ) )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1548_UnCI,axiom,
    ! [C: real,B4: set_real,A4: set_real] :
      ( ( ~ ( member_real @ C @ B4 )
       => ( member_real @ C @ A4 ) )
     => ( member_real @ C @ ( sup_sup_set_real @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1549_UnCI,axiom,
    ! [C: $o,B4: set_o,A4: set_o] :
      ( ( ~ ( member_o @ C @ B4 )
       => ( member_o @ C @ A4 ) )
     => ( member_o @ C @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1550_UnCI,axiom,
    ! [C: int,B4: set_int,A4: set_int] :
      ( ( ~ ( member_int @ C @ B4 )
       => ( member_int @ C @ A4 ) )
     => ( member_int @ C @ ( sup_sup_set_int @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1551_UnCI,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ A4 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1552_UnCI,axiom,
    ! [C: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ C @ B4 )
       => ( member8757157785044589968at_nat @ C @ A4 ) )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_1553_Un__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A4 @ B4 ) )
      = ( ( member_complex @ C @ A4 )
        | ( member_complex @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1554_Un__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A4 @ B4 ) )
      = ( ( member_real @ C @ A4 )
        | ( member_real @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1555_Un__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A4 @ B4 ) )
      = ( ( member_o @ C @ A4 )
        | ( member_o @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1556_Un__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A4 @ B4 ) )
      = ( ( member_int @ C @ A4 )
        | ( member_int @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1557_Un__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        | ( member_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1558_Un__iff,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( ( member8757157785044589968at_nat @ C @ A4 )
        | ( member8757157785044589968at_nat @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_1559_semiring__norm_I83_J,axiom,
    ! [N: num] :
      ( one
     != ( bit0 @ N ) ) ).

% semiring_norm(83)
thf(fact_1560_semiring__norm_I85_J,axiom,
    ! [M2: num] :
      ( ( bit0 @ M2 )
     != one ) ).

% semiring_norm(85)
thf(fact_1561_Un__empty,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ( A4 = bot_bo228742789529271731at_nat )
        & ( B4 = bot_bo228742789529271731at_nat ) ) ) ).

% Un_empty
thf(fact_1562_Un__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ( A4 = bot_bo2099793752762293965at_nat )
        & ( B4 = bot_bo2099793752762293965at_nat ) ) ) ).

% Un_empty
thf(fact_1563_Un__empty,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( sup_sup_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ( A4 = bot_bot_set_o )
        & ( B4 = bot_bot_set_o ) ) ) ).

% Un_empty
thf(fact_1564_Un__empty,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( sup_sup_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ( A4 = bot_bot_set_nat )
        & ( B4 = bot_bot_set_nat ) ) ) ).

% Un_empty
thf(fact_1565_Un__empty,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( sup_sup_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ( A4 = bot_bot_set_int )
        & ( B4 = bot_bot_set_int ) ) ) ).

% Un_empty
thf(fact_1566_Un__subset__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 )
      = ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
        & ( ord_le1268244103169919719at_nat @ B4 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1567_Un__subset__iff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
      = ( ( ord_less_eq_set_nat @ A4 @ C2 )
        & ( ord_less_eq_set_nat @ B4 @ C2 ) ) ) ).

% Un_subset_iff
thf(fact_1568_Un__insert__right,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( sup_sup_set_int @ A4 @ ( insert_int @ A @ B4 ) )
      = ( insert_int @ A @ ( sup_sup_set_int @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1569_Un__insert__right,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( sup_sup_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( insert_o @ A @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1570_Un__insert__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1571_Un__insert__right,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1572_Un__insert__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% Un_insert_right
thf(fact_1573_Un__insert__left,axiom,
    ! [A: int,B4: set_int,C2: set_int] :
      ( ( sup_sup_set_int @ ( insert_int @ A @ B4 ) @ C2 )
      = ( insert_int @ A @ ( sup_sup_set_int @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1574_Un__insert__left,axiom,
    ! [A: $o,B4: set_o,C2: set_o] :
      ( ( sup_sup_set_o @ ( insert_o @ A @ B4 ) @ C2 )
      = ( insert_o @ A @ ( sup_sup_set_o @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1575_Un__insert__left,axiom,
    ! [A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
      = ( insert8211810215607154385at_nat @ A @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1576_Un__insert__left,axiom,
    ! [A: nat,B4: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
      = ( insert_nat @ A @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1577_Un__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
      = ( insert9069300056098147895at_nat @ A @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) ) ).

% Un_insert_left
thf(fact_1578_semiring__norm_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(6)
thf(fact_1579_psubsetD,axiom,
    ! [A4: set_complex,B4: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A4 @ B4 )
     => ( ( member_complex @ C @ A4 )
       => ( member_complex @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1580_psubsetD,axiom,
    ! [A4: set_real,B4: set_real,C: real] :
      ( ( ord_less_set_real @ A4 @ B4 )
     => ( ( member_real @ C @ A4 )
       => ( member_real @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1581_psubsetD,axiom,
    ! [A4: set_o,B4: set_o,C: $o] :
      ( ( ord_less_set_o @ A4 @ B4 )
     => ( ( member_o @ C @ A4 )
       => ( member_o @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1582_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1583_psubsetD,axiom,
    ! [A4: set_int,B4: set_int,C: int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ( ( member_int @ C @ A4 )
       => ( member_int @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_1584_UnE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ~ ( member_complex @ C @ A4 )
       => ( member_complex @ C @ B4 ) ) ) ).

% UnE
thf(fact_1585_UnE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( sup_sup_set_real @ A4 @ B4 ) )
     => ( ~ ( member_real @ C @ A4 )
       => ( member_real @ C @ B4 ) ) ) ).

% UnE
thf(fact_1586_UnE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( sup_sup_set_o @ A4 @ B4 ) )
     => ( ~ ( member_o @ C @ A4 )
       => ( member_o @ C @ B4 ) ) ) ).

% UnE
thf(fact_1587_UnE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( sup_sup_set_int @ A4 @ B4 ) )
     => ( ~ ( member_int @ C @ A4 )
       => ( member_int @ C @ B4 ) ) ) ).

% UnE
thf(fact_1588_UnE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ~ ( member_nat @ C @ A4 )
       => ( member_nat @ C @ B4 ) ) ) ).

% UnE
thf(fact_1589_UnE,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
     => ( ~ ( member8757157785044589968at_nat @ C @ A4 )
       => ( member8757157785044589968at_nat @ C @ B4 ) ) ) ).

% UnE
thf(fact_1590_UnI1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ A4 )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1591_UnI1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ A4 )
     => ( member_real @ C @ ( sup_sup_set_real @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1592_UnI1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ A4 )
     => ( member_o @ C @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1593_UnI1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ A4 )
     => ( member_int @ C @ ( sup_sup_set_int @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1594_UnI1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1595_UnI1,axiom,
    ! [C: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ A4 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_1596_UnI2,axiom,
    ! [C: complex,B4: set_complex,A4: set_complex] :
      ( ( member_complex @ C @ B4 )
     => ( member_complex @ C @ ( sup_sup_set_complex @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1597_UnI2,axiom,
    ! [C: real,B4: set_real,A4: set_real] :
      ( ( member_real @ C @ B4 )
     => ( member_real @ C @ ( sup_sup_set_real @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1598_UnI2,axiom,
    ! [C: $o,B4: set_o,A4: set_o] :
      ( ( member_o @ C @ B4 )
     => ( member_o @ C @ ( sup_sup_set_o @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1599_UnI2,axiom,
    ! [C: int,B4: set_int,A4: set_int] :
      ( ( member_int @ C @ B4 )
     => ( member_int @ C @ ( sup_sup_set_int @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1600_UnI2,axiom,
    ! [C: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ C @ B4 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1601_UnI2,axiom,
    ! [C: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ C @ B4 )
     => ( member8757157785044589968at_nat @ C @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_1602_bex__Un,axiom,
    ! [A4: set_nat,B4: set_nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A4 @ B4 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P @ X4 ) )
        | ? [X4: nat] :
            ( ( member_nat @ X4 @ B4 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_1603_bex__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
            & ( P @ X4 ) ) )
      = ( ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A4 )
            & ( P @ X4 ) )
        | ? [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B4 )
            & ( P @ X4 ) ) ) ) ).

% bex_Un
thf(fact_1604_ball__Un,axiom,
    ! [A4: set_nat,B4: set_nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( sup_sup_set_nat @ A4 @ B4 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ( P @ X4 ) )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ B4 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_1605_ball__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
           => ( P @ X4 ) ) )
      = ( ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ A4 )
           => ( P @ X4 ) )
        & ! [X4: produc3843707927480180839at_nat] :
            ( ( member8757157785044589968at_nat @ X4 @ B4 )
           => ( P @ X4 ) ) ) ) ).

% ball_Un
thf(fact_1606_Un__assoc,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
      = ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).

% Un_assoc
thf(fact_1607_Un__assoc,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) ) ).

% Un_assoc
thf(fact_1608_Un__absorb,axiom,
    ! [A4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_1609_Un__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_1610_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_1611_Un__commute,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] : ( sup_su5525570899277871387at_nat @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_1612_Un__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( sup_sup_set_nat @ A4 @ B4 ) ) ).

% Un_left_absorb
thf(fact_1613_Un__left__absorb,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_left_absorb
thf(fact_1614_Un__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
      = ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A4 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_1615_Un__left__commute,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ B4 @ ( sup_su5525570899277871387at_nat @ A4 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_1616_Un__empty__right,axiom,
    ! [A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ bot_bo228742789529271731at_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1617_Un__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1618_Un__empty__right,axiom,
    ! [A4: set_o] :
      ( ( sup_sup_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Un_empty_right
thf(fact_1619_Un__empty__right,axiom,
    ! [A4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Un_empty_right
thf(fact_1620_Un__empty__right,axiom,
    ! [A4: set_int] :
      ( ( sup_sup_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Un_empty_right
thf(fact_1621_Un__empty__left,axiom,
    ! [B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1622_Un__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1623_Un__empty__left,axiom,
    ! [B4: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1624_Un__empty__left,axiom,
    ! [B4: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1625_Un__empty__left,axiom,
    ! [B4: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
      = B4 ) ).

% Un_empty_left
thf(fact_1626_subset__Un__eq,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_1627_subset__Un__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B5 )
          = B5 ) ) ) ).

% subset_Un_eq
thf(fact_1628_subset__UnE,axiom,
    ! [C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C2 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
     => ~ ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( ord_le1268244103169919719at_nat @ A7 @ A4 )
           => ! [B8: set_Pr4329608150637261639at_nat] :
                ( ( ord_le1268244103169919719at_nat @ B8 @ B4 )
               => ( C2
                 != ( sup_su5525570899277871387at_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_1629_subset__UnE,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ~ ! [A7: set_nat] :
            ( ( ord_less_eq_set_nat @ A7 @ A4 )
           => ! [B8: set_nat] :
                ( ( ord_less_eq_set_nat @ B8 @ B4 )
               => ( C2
                 != ( sup_sup_set_nat @ A7 @ B8 ) ) ) ) ) ).

% subset_UnE
thf(fact_1630_Un__absorb2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B4 @ A4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_1631_Un__absorb2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( sup_sup_set_nat @ A4 @ B4 )
        = A4 ) ) ).

% Un_absorb2
thf(fact_1632_Un__absorb1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_1633_Un__absorb1,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( sup_sup_set_nat @ A4 @ B4 )
        = B4 ) ) ).

% Un_absorb1
thf(fact_1634_Un__upper2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_upper2
thf(fact_1635_Un__upper2,axiom,
    ! [B4: set_nat,A4: set_nat] : ( ord_less_eq_set_nat @ B4 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ).

% Un_upper2
thf(fact_1636_Un__upper1,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_upper1
thf(fact_1637_Un__upper1,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ).

% Un_upper1
thf(fact_1638_Un__least,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ C2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 ) ) ) ).

% Un_least
thf(fact_1639_Un__least,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 ) ) ) ).

% Un_least
thf(fact_1640_Un__mono,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,D2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ C2 )
     => ( ( ord_le1268244103169919719at_nat @ B4 @ D2 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ ( sup_su5525570899277871387at_nat @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_1641_Un__mono,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ B4 @ D2 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).

% Un_mono
thf(fact_1642_singleton__Un__iff,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat )
        = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1643_singleton__Un__iff,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat )
        = ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1644_singleton__Un__iff,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ( ( insert_o @ X3 @ bot_bot_set_o )
        = ( sup_sup_set_o @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bot_set_o )
          & ( B4
            = ( insert_o @ X3 @ bot_bot_set_o ) ) )
        | ( ( A4
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B4
            = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1645_singleton__Un__iff,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ( ( insert_nat @ X3 @ bot_bot_set_nat )
        = ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bot_set_nat )
          & ( B4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
        | ( ( A4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1646_singleton__Un__iff,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ( ( insert_int @ X3 @ bot_bot_set_int )
        = ( sup_sup_set_int @ A4 @ B4 ) )
      = ( ( ( A4 = bot_bot_set_int )
          & ( B4
            = ( insert_int @ X3 @ bot_bot_set_int ) ) )
        | ( ( A4
            = ( insert_int @ X3 @ bot_bot_set_int ) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4
            = ( insert_int @ X3 @ bot_bot_set_int ) )
          & ( B4
            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1647_Un__singleton__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A4 @ B4 )
        = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
      = ( ( ( A4 = bot_bo228742789529271731at_nat )
          & ( B4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
          & ( B4 = bot_bo228742789529271731at_nat ) )
        | ( ( A4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
          & ( B4
            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1648_Un__singleton__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A4 @ B4 )
        = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
      = ( ( ( A4 = bot_bo2099793752762293965at_nat )
          & ( B4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
          & ( B4 = bot_bo2099793752762293965at_nat ) )
        | ( ( A4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
          & ( B4
            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1649_Un__singleton__iff,axiom,
    ! [A4: set_o,B4: set_o,X3: $o] :
      ( ( ( sup_sup_set_o @ A4 @ B4 )
        = ( insert_o @ X3 @ bot_bot_set_o ) )
      = ( ( ( A4 = bot_bot_set_o )
          & ( B4
            = ( insert_o @ X3 @ bot_bot_set_o ) ) )
        | ( ( A4
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B4 = bot_bot_set_o ) )
        | ( ( A4
            = ( insert_o @ X3 @ bot_bot_set_o ) )
          & ( B4
            = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1650_Un__singleton__iff,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( ( ( sup_sup_set_nat @ A4 @ B4 )
        = ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = ( ( ( A4 = bot_bot_set_nat )
          & ( B4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
        | ( ( A4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B4 = bot_bot_set_nat ) )
        | ( ( A4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) )
          & ( B4
            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1651_Un__singleton__iff,axiom,
    ! [A4: set_int,B4: set_int,X3: int] :
      ( ( ( sup_sup_set_int @ A4 @ B4 )
        = ( insert_int @ X3 @ bot_bot_set_int ) )
      = ( ( ( A4 = bot_bot_set_int )
          & ( B4
            = ( insert_int @ X3 @ bot_bot_set_int ) ) )
        | ( ( A4
            = ( insert_int @ X3 @ bot_bot_set_int ) )
          & ( B4 = bot_bot_set_int ) )
        | ( ( A4
            = ( insert_int @ X3 @ bot_bot_set_int ) )
          & ( B4
            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1652_insert__is__Un,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A6: produc3843707927480180839at_nat] : ( sup_su5525570899277871387at_nat @ ( insert9069300056098147895at_nat @ A6 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1653_insert__is__Un,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A6: product_prod_nat_nat] : ( sup_su6327502436637775413at_nat @ ( insert8211810215607154385at_nat @ A6 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1654_insert__is__Un,axiom,
    ( insert_o
    = ( ^ [A6: $o] : ( sup_sup_set_o @ ( insert_o @ A6 @ bot_bot_set_o ) ) ) ) ).

% insert_is_Un
thf(fact_1655_insert__is__Un,axiom,
    ( insert_nat
    = ( ^ [A6: nat] : ( sup_sup_set_nat @ ( insert_nat @ A6 @ bot_bot_set_nat ) ) ) ) ).

% insert_is_Un
thf(fact_1656_insert__is__Un,axiom,
    ( insert_int
    = ( ^ [A6: int] : ( sup_sup_set_int @ ( insert_int @ A6 @ bot_bot_set_int ) ) ) ) ).

% insert_is_Un
thf(fact_1657_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X3 ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_1658_case4_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 @ treeList2 @ I2 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X: nat] :
              ( ( ( ( vEBT_VEBT_high @ X @ na )
                  = I2 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ na ) ) )
             => ( ( ord_less_nat @ mi @ X )
                & ( ord_less_eq_nat @ X @ ma ) ) ) ) ) ) ).

% case4(11)
thf(fact_1659_enat__ord__number_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(1)
thf(fact_1660_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A @ bot_bo228742789529271731at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1661_sup__bot_Oright__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A @ bot_bo2099793752762293965at_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1662_sup__bot_Oright__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ A @ bot_bot_set_o )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1663_sup__bot_Oright__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1664_sup__bot_Oright__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ A @ bot_bot_set_int )
      = A ) ).

% sup_bot.right_neutral
thf(fact_1665_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1666_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ A @ B ) )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1667_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ A @ B ) )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1668_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ A @ B ) )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1669_sup__bot_Oneutr__eq__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ A @ B ) )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.neutr_eq_iff
thf(fact_1670_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1671_sup__bot_Oleft__neutral,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1672_sup__bot_Oleft__neutral,axiom,
    ! [A: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1673_sup__bot_Oleft__neutral,axiom,
    ! [A: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1674_sup__bot_Oleft__neutral,axiom,
    ! [A: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ A )
      = A ) ).

% sup_bot.left_neutral
thf(fact_1675_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ A @ B )
        = bot_bo228742789529271731at_nat )
      = ( ( A = bot_bo228742789529271731at_nat )
        & ( B = bot_bo228742789529271731at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1676_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ A @ B )
        = bot_bo2099793752762293965at_nat )
      = ( ( A = bot_bo2099793752762293965at_nat )
        & ( B = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1677_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_o,B: set_o] :
      ( ( ( sup_sup_set_o @ A @ B )
        = bot_bot_set_o )
      = ( ( A = bot_bot_set_o )
        & ( B = bot_bot_set_o ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1678_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( sup_sup_set_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ( A = bot_bot_set_nat )
        & ( B = bot_bot_set_nat ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1679_sup__bot_Oeq__neutr__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( sup_sup_set_int @ A @ B )
        = bot_bot_set_int )
      = ( ( A = bot_bot_set_int )
        & ( B = bot_bot_set_int ) ) ) ).

% sup_bot.eq_neutr_iff
thf(fact_1680_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ X3 @ Y )
        = bot_bo228742789529271731at_nat )
      = ( ( X3 = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_1681_sup__eq__bot__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ X3 @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ( X3 = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_1682_sup__eq__bot__iff,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( ( sup_sup_set_o @ X3 @ Y )
        = bot_bot_set_o )
      = ( ( X3 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% sup_eq_bot_iff
thf(fact_1683_sup__eq__bot__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ( sup_sup_set_nat @ X3 @ Y )
        = bot_bot_set_nat )
      = ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% sup_eq_bot_iff
thf(fact_1684_sup__eq__bot__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( ( sup_sup_set_int @ X3 @ Y )
        = bot_bot_set_int )
      = ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% sup_eq_bot_iff
thf(fact_1685_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( sup_su5525570899277871387at_nat @ X3 @ Y ) )
      = ( ( X3 = bot_bo228742789529271731at_nat )
        & ( Y = bot_bo228742789529271731at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_1686_bot__eq__sup__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( sup_su6327502436637775413at_nat @ X3 @ Y ) )
      = ( ( X3 = bot_bo2099793752762293965at_nat )
        & ( Y = bot_bo2099793752762293965at_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_1687_bot__eq__sup__iff,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( bot_bot_set_o
        = ( sup_sup_set_o @ X3 @ Y ) )
      = ( ( X3 = bot_bot_set_o )
        & ( Y = bot_bot_set_o ) ) ) ).

% bot_eq_sup_iff
thf(fact_1688_bot__eq__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( bot_bot_set_nat
        = ( sup_sup_set_nat @ X3 @ Y ) )
      = ( ( X3 = bot_bot_set_nat )
        & ( Y = bot_bot_set_nat ) ) ) ).

% bot_eq_sup_iff
thf(fact_1689_bot__eq__sup__iff,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( bot_bot_set_int
        = ( sup_sup_set_int @ X3 @ Y ) )
      = ( ( X3 = bot_bot_set_int )
        & ( Y = bot_bot_set_int ) ) ) ).

% bot_eq_sup_iff
thf(fact_1690_sup__bot__right,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X3 @ bot_bo228742789529271731at_nat )
      = X3 ) ).

% sup_bot_right
thf(fact_1691_sup__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      = X3 ) ).

% sup_bot_right
thf(fact_1692_sup__bot__right,axiom,
    ! [X3: set_o] :
      ( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
      = X3 ) ).

% sup_bot_right
thf(fact_1693_sup__bot__right,axiom,
    ! [X3: set_nat] :
      ( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
      = X3 ) ).

% sup_bot_right
thf(fact_1694_sup__bot__right,axiom,
    ! [X3: set_int] :
      ( ( sup_sup_set_int @ X3 @ bot_bot_set_int )
      = X3 ) ).

% sup_bot_right
thf(fact_1695_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_1696_Diff__idemp,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ B4 )
      = ( minus_minus_set_nat @ A4 @ B4 ) ) ).

% Diff_idemp
thf(fact_1697_Diff__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
      = ( ( member_complex @ C @ A4 )
        & ~ ( member_complex @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1698_Diff__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
      = ( ( member_real @ C @ A4 )
        & ~ ( member_real @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1699_Diff__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A4 @ B4 ) )
      = ( ( member_o @ C @ A4 )
        & ~ ( member_o @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1700_Diff__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
      = ( ( member_int @ C @ A4 )
        & ~ ( member_int @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1701_Diff__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        & ~ ( member_nat @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_1702_DiffI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ A4 )
     => ( ~ ( member_complex @ C @ B4 )
       => ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1703_DiffI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ A4 )
     => ( ~ ( member_real @ C @ B4 )
       => ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1704_DiffI,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ A4 )
     => ( ~ ( member_o @ C @ B4 )
       => ( member_o @ C @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1705_DiffI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ A4 )
     => ( ~ ( member_int @ C @ B4 )
       => ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1706_DiffI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( ~ ( member_nat @ C @ B4 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ).

% DiffI
thf(fact_1707_high__bound__aux,axiom,
    ! [Ma: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% high_bound_aux
thf(fact_1708_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X3 ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_1709_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1710_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1711_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1712_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1713_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1714_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1715_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1716_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1717_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1718_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1719_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1720_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1721_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1722_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_1723_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1724_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1725_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1726_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1727_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1728_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1729_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1730_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_1731_le__sup__iff,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) @ Z2 )
      = ( ( ord_le1268244103169919719at_nat @ X3 @ Z2 )
        & ( ord_le1268244103169919719at_nat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1732_le__sup__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ Z2 )
      = ( ( ord_less_eq_set_nat @ X3 @ Z2 )
        & ( ord_less_eq_set_nat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1733_le__sup__iff,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X3 @ Y ) @ Z2 )
      = ( ( ord_less_eq_rat @ X3 @ Z2 )
        & ( ord_less_eq_rat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1734_le__sup__iff,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ Y ) @ Z2 )
      = ( ( ord_less_eq_nat @ X3 @ Z2 )
        & ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1735_le__sup__iff,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X3 @ Y ) @ Z2 )
      = ( ( ord_less_eq_int @ X3 @ Z2 )
        & ( ord_less_eq_int @ Y @ Z2 ) ) ) ).

% le_sup_iff
thf(fact_1736_sup_Obounded__iff,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
      = ( ( ord_le1268244103169919719at_nat @ B @ A )
        & ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1737_sup_Obounded__iff,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1738_sup_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1739_sup_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1740_sup_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.bounded_iff
thf(fact_1741_sup__bot__left,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ bot_bo228742789529271731at_nat @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1742_sup__bot__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ bot_bo2099793752762293965at_nat @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1743_sup__bot__left,axiom,
    ! [X3: set_o] :
      ( ( sup_sup_set_o @ bot_bot_set_o @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1744_sup__bot__left,axiom,
    ! [X3: set_nat] :
      ( ( sup_sup_set_nat @ bot_bot_set_nat @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1745_sup__bot__left,axiom,
    ! [X3: set_int] :
      ( ( sup_sup_set_int @ bot_bot_set_int @ X3 )
      = X3 ) ).

% sup_bot_left
thf(fact_1746_Diff__empty,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = A4 ) ).

% Diff_empty
thf(fact_1747_Diff__empty,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ bot_bot_set_o )
      = A4 ) ).

% Diff_empty
thf(fact_1748_Diff__empty,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ bot_bot_set_int )
      = A4 ) ).

% Diff_empty
thf(fact_1749_Diff__empty,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ bot_bot_set_nat )
      = A4 ) ).

% Diff_empty
thf(fact_1750_empty__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_1751_empty__Diff,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ bot_bot_set_o @ A4 )
      = bot_bot_set_o ) ).

% empty_Diff
thf(fact_1752_empty__Diff,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A4 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_1753_empty__Diff,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A4 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_1754_Diff__cancel,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_1755_Diff__cancel,axiom,
    ! [A4: set_o] :
      ( ( minus_minus_set_o @ A4 @ A4 )
      = bot_bot_set_o ) ).

% Diff_cancel
thf(fact_1756_Diff__cancel,axiom,
    ! [A4: set_int] :
      ( ( minus_minus_set_int @ A4 @ A4 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_1757_Diff__cancel,axiom,
    ! [A4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ A4 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_1758_Diff__insert0,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ B4 ) )
        = ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1759_Diff__insert0,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
     => ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ B4 ) )
        = ( minus_3314409938677909166at_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1760_Diff__insert0,axiom,
    ! [X3: complex,A4: set_complex,B4: set_complex] :
      ( ~ ( member_complex @ X3 @ A4 )
     => ( ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ B4 ) )
        = ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1761_Diff__insert0,axiom,
    ! [X3: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1762_Diff__insert0,axiom,
    ! [X3: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1763_Diff__insert0,axiom,
    ! [X3: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ X3 @ A4 )
     => ( ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ B4 ) )
        = ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1764_Diff__insert0,axiom,
    ! [X3: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% Diff_insert0
thf(fact_1765_insert__Diff1,axiom,
    ! [X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ B4 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) @ B4 )
        = ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1766_insert__Diff1,axiom,
    ! [X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ B4 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) @ B4 )
        = ( minus_3314409938677909166at_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1767_insert__Diff1,axiom,
    ! [X3: complex,B4: set_complex,A4: set_complex] :
      ( ( member_complex @ X3 @ B4 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X3 @ A4 ) @ B4 )
        = ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1768_insert__Diff1,axiom,
    ! [X3: real,B4: set_real,A4: set_real] :
      ( ( member_real @ X3 @ B4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_real @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1769_insert__Diff1,axiom,
    ! [X3: $o,B4: set_o,A4: set_o] :
      ( ( member_o @ X3 @ B4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1770_insert__Diff1,axiom,
    ! [X3: int,B4: set_int,A4: set_int] :
      ( ( member_int @ X3 @ B4 )
     => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1771_insert__Diff1,axiom,
    ! [X3: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ X3 @ B4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
        = ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% insert_Diff1
thf(fact_1772_Un__Diff__cancel2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ B4 @ A4 ) @ A4 )
      = ( sup_su5525570899277871387at_nat @ B4 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_1773_Un__Diff__cancel2,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B4 @ A4 ) @ A4 )
      = ( sup_sup_set_nat @ B4 @ A4 ) ) ).

% Un_Diff_cancel2
thf(fact_1774_Un__Diff__cancel,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B4 @ A4 ) )
      = ( sup_su5525570899277871387at_nat @ A4 @ B4 ) ) ).

% Un_Diff_cancel
thf(fact_1775_Un__Diff__cancel,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
      = ( sup_sup_set_nat @ A4 @ B4 ) ) ).

% Un_Diff_cancel
thf(fact_1776_Diff__eq__empty__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1777_Diff__eq__empty__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( minus_minus_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1778_Diff__eq__empty__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( minus_minus_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1779_Diff__eq__empty__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( minus_minus_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% Diff_eq_empty_iff
thf(fact_1780_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1781_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1782_insert__Diff__single,axiom,
    ! [A: $o,A4: set_o] :
      ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
      = ( insert_o @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1783_insert__Diff__single,axiom,
    ! [A: int,A4: set_int] :
      ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
      = ( insert_int @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1784_insert__Diff__single,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
      = ( insert_nat @ A @ A4 ) ) ).

% insert_Diff_single
thf(fact_1785_enat__ord__number_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% enat_ord_number(2)
thf(fact_1786_acd,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 ) ) )
          & ! [X: nat] :
              ( ( ( ( vEBT_VEBT_high @ X @ na )
                  = I2 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I2 ) @ ( vEBT_VEBT_low @ X @ na ) ) )
             => ( ( ord_less_nat @ mi @ X )
                & ( ord_less_eq_nat @ X @ ma ) ) ) ) ) ) ).

% acd
thf(fact_1787_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1788_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1789_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1790_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_1791_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1792_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1793_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1794_DiffD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ~ ( member_complex @ C @ B4 ) ) ).

% DiffD2
thf(fact_1795_DiffD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( member_real @ C @ B4 ) ) ).

% DiffD2
thf(fact_1796_DiffD2,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A4 @ B4 ) )
     => ~ ( member_o @ C @ B4 ) ) ).

% DiffD2
thf(fact_1797_DiffD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( member_int @ C @ B4 ) ) ).

% DiffD2
thf(fact_1798_DiffD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( member_nat @ C @ B4 ) ) ).

% DiffD2
thf(fact_1799_DiffD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ( member_complex @ C @ A4 ) ) ).

% DiffD1
thf(fact_1800_DiffD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ( member_real @ C @ A4 ) ) ).

% DiffD1
thf(fact_1801_DiffD1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A4 @ B4 ) )
     => ( member_o @ C @ A4 ) ) ).

% DiffD1
thf(fact_1802_DiffD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ( member_int @ C @ A4 ) ) ).

% DiffD1
thf(fact_1803_DiffD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C @ A4 ) ) ).

% DiffD1
thf(fact_1804_DiffE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
     => ~ ( ( member_complex @ C @ A4 )
         => ( member_complex @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1805_DiffE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A4 @ B4 ) )
     => ~ ( ( member_real @ C @ A4 )
         => ( member_real @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1806_DiffE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( minus_minus_set_o @ A4 @ B4 ) )
     => ~ ( ( member_o @ C @ A4 )
         => ( member_o @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1807_DiffE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A4 @ B4 ) )
     => ~ ( ( member_int @ C @ A4 )
         => ( member_int @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1808_DiffE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ C @ A4 )
         => ( member_nat @ C @ B4 ) ) ) ).

% DiffE
thf(fact_1809_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1810_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1811_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1812_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1813_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1814_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_1815_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1816_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1817_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_1818_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1819_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1820_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1821_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1822_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1823_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_1824_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1825_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1826_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1827_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1828_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1829_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1830_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1831_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1832_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1833_group__cancel_Osub1,axiom,
    ! [A4: real,K2: real,A: real,B: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( minus_minus_real @ A4 @ B )
        = ( plus_plus_real @ K2 @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1834_group__cancel_Osub1,axiom,
    ! [A4: rat,K2: rat,A: rat,B: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( minus_minus_rat @ A4 @ B )
        = ( plus_plus_rat @ K2 @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1835_group__cancel_Osub1,axiom,
    ! [A4: int,K2: int,A: int,B: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( minus_minus_int @ A4 @ B )
        = ( plus_plus_int @ K2 @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_1836_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1837_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1838_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_1839_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1840_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1841_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_1842_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1843_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1844_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_1845_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1846_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1847_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1848_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1849_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1850_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_1851_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1852_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1853_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1854_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1855_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1856_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1857_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1858_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1859_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1860_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1861_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_1862_Diff__mono,axiom,
    ! [A4: set_nat,C2: set_nat,D2: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ D2 @ B4 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ C2 @ D2 ) ) ) ) ).

% Diff_mono
thf(fact_1863_Diff__subset,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ A4 ) ).

% Diff_subset
thf(fact_1864_double__diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ C2 )
       => ( ( minus_minus_set_nat @ B4 @ ( minus_minus_set_nat @ C2 @ A4 ) )
          = A4 ) ) ) ).

% double_diff
thf(fact_1865_insert__Diff__if,axiom,
    ! [X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X3 @ B4 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) @ B4 )
          = ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X3 @ B4 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) @ B4 )
          = ( insert8211810215607154385at_nat @ X3 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1866_insert__Diff__if,axiom,
    ! [X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X3 @ B4 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) @ B4 )
          = ( minus_3314409938677909166at_nat @ A4 @ B4 ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X3 @ B4 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) @ B4 )
          = ( insert9069300056098147895at_nat @ X3 @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1867_insert__Diff__if,axiom,
    ! [X3: complex,B4: set_complex,A4: set_complex] :
      ( ( ( member_complex @ X3 @ B4 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X3 @ A4 ) @ B4 )
          = ( minus_811609699411566653omplex @ A4 @ B4 ) ) )
      & ( ~ ( member_complex @ X3 @ B4 )
       => ( ( minus_811609699411566653omplex @ ( insert_complex @ X3 @ A4 ) @ B4 )
          = ( insert_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1868_insert__Diff__if,axiom,
    ! [X3: real,B4: set_real,A4: set_real] :
      ( ( ( member_real @ X3 @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_real @ A4 @ B4 ) ) )
      & ( ~ ( member_real @ X3 @ B4 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ B4 )
          = ( insert_real @ X3 @ ( minus_minus_set_real @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1869_insert__Diff__if,axiom,
    ! [X3: $o,B4: set_o,A4: set_o] :
      ( ( ( member_o @ X3 @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_o @ A4 @ B4 ) ) )
      & ( ~ ( member_o @ X3 @ B4 )
       => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ B4 )
          = ( insert_o @ X3 @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1870_insert__Diff__if,axiom,
    ! [X3: int,B4: set_int,A4: set_int] :
      ( ( ( member_int @ X3 @ B4 )
       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_int @ A4 @ B4 ) ) )
      & ( ~ ( member_int @ X3 @ B4 )
       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A4 ) @ B4 )
          = ( insert_int @ X3 @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1871_insert__Diff__if,axiom,
    ! [X3: nat,B4: set_nat,A4: set_nat] :
      ( ( ( member_nat @ X3 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
          = ( minus_minus_set_nat @ A4 @ B4 ) ) )
      & ( ~ ( member_nat @ X3 @ B4 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ B4 )
          = ( insert_nat @ X3 @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1872_Un__Diff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ C2 )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ C2 ) @ ( minus_3314409938677909166at_nat @ B4 @ C2 ) ) ) ).

% Un_Diff
thf(fact_1873_Un__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ C2 ) @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).

% Un_Diff
thf(fact_1874_psubset__imp__ex__mem,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( ord_less_set_complex @ A4 @ B4 )
     => ? [B3: complex] : ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1875_psubset__imp__ex__mem,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ord_less_set_real @ A4 @ B4 )
     => ? [B3: real] : ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1876_psubset__imp__ex__mem,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ord_less_set_o @ A4 @ B4 )
     => ? [B3: $o] : ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1877_psubset__imp__ex__mem,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ B4 )
     => ? [B3: int] : ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1878_psubset__imp__ex__mem,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1879_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1880_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1881_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_1882_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1883_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1884_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_1885_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_1886_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_1887_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1888_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1889_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1890_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1891_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1892_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1893_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1894_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1895_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1896_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1897_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_1898_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1899_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1900_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_1901_Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_1902_Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_1903_Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ B4 ) @ ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% Diff_insert
thf(fact_1904_Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ B4 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_1905_Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_1906_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1907_insert__Diff,axiom,
    ! [A: complex,A4: set_complex] :
      ( ( member_complex @ A @ A4 )
     => ( ( insert_complex @ A @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1908_insert__Diff,axiom,
    ! [A: real,A4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( insert_real @ A @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1909_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1910_insert__Diff,axiom,
    ! [A: $o,A4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( insert_o @ A @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1911_insert__Diff,axiom,
    ! [A: int,A4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( insert_int @ A @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1912_insert__Diff,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
        = A4 ) ) ).

% insert_Diff
thf(fact_1913_Diff__insert2,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1914_Diff__insert2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1915_Diff__insert2,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) )
      = ( minus_minus_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1916_Diff__insert2,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1917_Diff__insert2,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_1918_Diff__insert__absorb,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1919_Diff__insert__absorb,axiom,
    ! [X3: complex,A4: set_complex] :
      ( ~ ( member_complex @ X3 @ A4 )
     => ( ( minus_811609699411566653omplex @ ( insert_complex @ X3 @ A4 ) @ ( insert_complex @ X3 @ bot_bot_set_complex ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1920_Diff__insert__absorb,axiom,
    ! [X3: real,A4: set_real] :
      ( ~ ( member_real @ X3 @ A4 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A4 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1921_Diff__insert__absorb,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1922_Diff__insert__absorb,axiom,
    ! [X3: $o,A4: set_o] :
      ( ~ ( member_o @ X3 @ A4 )
     => ( ( minus_minus_set_o @ ( insert_o @ X3 @ A4 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1923_Diff__insert__absorb,axiom,
    ! [X3: int,A4: set_int] :
      ( ~ ( member_int @ X3 @ A4 )
     => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A4 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1924_Diff__insert__absorb,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ~ ( member_nat @ X3 @ A4 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A4 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
        = A4 ) ) ).

% Diff_insert_absorb
thf(fact_1925_subset__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B4 @ ( insert8211810215607154385at_nat @ X3 @ C2 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B4 @ C2 ) )
        & ~ ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1926_subset__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B4 @ ( insert9069300056098147895at_nat @ X3 @ C2 ) ) )
      = ( ( ord_le1268244103169919719at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B4 @ C2 ) )
        & ~ ( member8757157785044589968at_nat @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1927_subset__Diff__insert,axiom,
    ! [A4: set_complex,B4: set_complex,X3: complex,C2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ ( minus_811609699411566653omplex @ B4 @ ( insert_complex @ X3 @ C2 ) ) )
      = ( ( ord_le211207098394363844omplex @ A4 @ ( minus_811609699411566653omplex @ B4 @ C2 ) )
        & ~ ( member_complex @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1928_subset__Diff__insert,axiom,
    ! [A4: set_real,B4: set_real,X3: real,C2: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ ( insert_real @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_real @ A4 @ ( minus_minus_set_real @ B4 @ C2 ) )
        & ~ ( member_real @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1929_subset__Diff__insert,axiom,
    ! [A4: set_o,B4: set_o,X3: $o,C2: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ ( insert_o @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_o @ A4 @ ( minus_minus_set_o @ B4 @ C2 ) )
        & ~ ( member_o @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1930_subset__Diff__insert,axiom,
    ! [A4: set_int,B4: set_int,X3: int,C2: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B4 @ ( insert_int @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_int @ A4 @ ( minus_minus_set_int @ B4 @ C2 ) )
        & ~ ( member_int @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1931_subset__Diff__insert,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X3 @ C2 ) ) )
      = ( ( ord_less_eq_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C2 ) )
        & ~ ( member_nat @ X3 @ A4 ) ) ) ).

% subset_Diff_insert
thf(fact_1932_Diff__subset__conv,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) @ C2 )
      = ( ord_le1268244103169919719at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_1933_Diff__subset__conv,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ C2 )
      = ( ord_less_eq_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).

% Diff_subset_conv
thf(fact_1934_Diff__partition,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ B4 )
     => ( ( sup_su5525570899277871387at_nat @ A4 @ ( minus_3314409938677909166at_nat @ B4 @ A4 ) )
        = B4 ) ) ).

% Diff_partition
thf(fact_1935_Diff__partition,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( sup_sup_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
        = B4 ) ) ).

% Diff_partition
thf(fact_1936_subset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ B4 ) )
      = ( ( ( member8757157785044589968at_nat @ X3 @ A4 )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B4 ) )
        & ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
         => ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1937_subset__insert__iff,axiom,
    ! [A4: set_complex,X3: complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ ( insert_complex @ X3 @ B4 ) )
      = ( ( ( member_complex @ X3 @ A4 )
         => ( ord_le211207098394363844omplex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) @ B4 ) )
        & ( ~ ( member_complex @ X3 @ A4 )
         => ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1938_subset__insert__iff,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( ( ord_less_eq_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
      = ( ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B4 ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1939_subset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ B4 ) )
      = ( ( ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
        & ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
         => ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1940_subset__insert__iff,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
      = ( ( ( member_o @ X3 @ A4 )
         => ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1941_subset__insert__iff,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ X3 @ B4 ) )
      = ( ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B4 ) )
        & ( ~ ( member_int @ X3 @ A4 )
         => ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1942_subset__insert__iff,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
      = ( ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B4 ) )
        & ( ~ ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_1943_Diff__single__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B4 )
     => ( ord_le1268244103169919719at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1944_Diff__single__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B4 )
     => ( ord_le3146513528884898305at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1945_Diff__single__insert,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 )
     => ( ord_less_eq_set_o @ A4 @ ( insert_o @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1946_Diff__single__insert,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B4 )
     => ( ord_less_eq_set_int @ A4 @ ( insert_int @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1947_Diff__single__insert,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B4 )
     => ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_1948_power2__commute,axiom,
    ! [X3: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1949_power2__commute,axiom,
    ! [X3: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1950_power2__commute,axiom,
    ! [X3: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1951_power2__commute,axiom,
    ! [X3: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_1952_psubset__insert__iff,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ B4 ) )
      = ( ( ( member8757157785044589968at_nat @ X3 @ B4 )
         => ( ord_le2604355607129572851at_nat @ A4 @ B4 ) )
        & ( ~ ( member8757157785044589968at_nat @ X3 @ B4 )
         => ( ( ( member8757157785044589968at_nat @ X3 @ A4 )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B4 ) )
            & ( ~ ( member8757157785044589968at_nat @ X3 @ A4 )
             => ( ord_le1268244103169919719at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1953_psubset__insert__iff,axiom,
    ! [A4: set_complex,X3: complex,B4: set_complex] :
      ( ( ord_less_set_complex @ A4 @ ( insert_complex @ X3 @ B4 ) )
      = ( ( ( member_complex @ X3 @ B4 )
         => ( ord_less_set_complex @ A4 @ B4 ) )
        & ( ~ ( member_complex @ X3 @ B4 )
         => ( ( ( member_complex @ X3 @ A4 )
             => ( ord_less_set_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) @ B4 ) )
            & ( ~ ( member_complex @ X3 @ A4 )
             => ( ord_le211207098394363844omplex @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1954_psubset__insert__iff,axiom,
    ! [A4: set_real,X3: real,B4: set_real] :
      ( ( ord_less_set_real @ A4 @ ( insert_real @ X3 @ B4 ) )
      = ( ( ( member_real @ X3 @ B4 )
         => ( ord_less_set_real @ A4 @ B4 ) )
        & ( ~ ( member_real @ X3 @ B4 )
         => ( ( ( member_real @ X3 @ A4 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B4 ) )
            & ( ~ ( member_real @ X3 @ A4 )
             => ( ord_less_eq_set_real @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1955_psubset__insert__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ B4 ) )
      = ( ( ( member8440522571783428010at_nat @ X3 @ B4 )
         => ( ord_le7866589430770878221at_nat @ A4 @ B4 ) )
        & ( ~ ( member8440522571783428010at_nat @ X3 @ B4 )
         => ( ( ( member8440522571783428010at_nat @ X3 @ A4 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B4 ) )
            & ( ~ ( member8440522571783428010at_nat @ X3 @ A4 )
             => ( ord_le3146513528884898305at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1956_psubset__insert__iff,axiom,
    ! [A4: set_o,X3: $o,B4: set_o] :
      ( ( ord_less_set_o @ A4 @ ( insert_o @ X3 @ B4 ) )
      = ( ( ( member_o @ X3 @ B4 )
         => ( ord_less_set_o @ A4 @ B4 ) )
        & ( ~ ( member_o @ X3 @ B4 )
         => ( ( ( member_o @ X3 @ A4 )
             => ( ord_less_set_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) @ B4 ) )
            & ( ~ ( member_o @ X3 @ A4 )
             => ( ord_less_eq_set_o @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1957_psubset__insert__iff,axiom,
    ! [A4: set_int,X3: int,B4: set_int] :
      ( ( ord_less_set_int @ A4 @ ( insert_int @ X3 @ B4 ) )
      = ( ( ( member_int @ X3 @ B4 )
         => ( ord_less_set_int @ A4 @ B4 ) )
        & ( ~ ( member_int @ X3 @ B4 )
         => ( ( ( member_int @ X3 @ A4 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B4 ) )
            & ( ~ ( member_int @ X3 @ A4 )
             => ( ord_less_eq_set_int @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1958_psubset__insert__iff,axiom,
    ! [A4: set_nat,X3: nat,B4: set_nat] :
      ( ( ord_less_set_nat @ A4 @ ( insert_nat @ X3 @ B4 ) )
      = ( ( ( member_nat @ X3 @ B4 )
         => ( ord_less_set_nat @ A4 @ B4 ) )
        & ( ~ ( member_nat @ X3 @ B4 )
         => ( ( ( member_nat @ X3 @ A4 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B4 ) )
            & ( ~ ( member_nat @ X3 @ A4 )
             => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_1959_inf__sup__ord_I4_J,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1960_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1961_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X3: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1962_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1963_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X3: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X3 @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_1964_inf__sup__ord_I3_J,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1965_inf__sup__ord_I3_J,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1966_inf__sup__ord_I3_J,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ X3 @ ( sup_sup_rat @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1967_inf__sup__ord_I3_J,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1968_inf__sup__ord_I3_J,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ X3 @ ( sup_sup_int @ X3 @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_1969_le__supE,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_le1268244103169919719at_nat @ A @ X3 )
         => ~ ( ord_le1268244103169919719at_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1970_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_set_nat @ A @ X3 )
         => ~ ( ord_less_eq_set_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1971_le__supE,axiom,
    ! [A: rat,B: rat,X3: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_rat @ A @ X3 )
         => ~ ( ord_less_eq_rat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1972_le__supE,axiom,
    ! [A: nat,B: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_nat @ A @ X3 )
         => ~ ( ord_less_eq_nat @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1973_le__supE,axiom,
    ! [A: int,B: int,X3: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X3 )
     => ~ ( ( ord_less_eq_int @ A @ X3 )
         => ~ ( ord_less_eq_int @ B @ X3 ) ) ) ).

% le_supE
thf(fact_1974_le__supI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ X3 )
     => ( ( ord_le1268244103169919719at_nat @ B @ X3 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1975_le__supI,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X3 )
     => ( ( ord_less_eq_set_nat @ B @ X3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1976_le__supI,axiom,
    ! [A: rat,X3: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X3 )
     => ( ( ord_less_eq_rat @ B @ X3 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1977_le__supI,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ( ord_less_eq_nat @ B @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1978_le__supI,axiom,
    ! [A: int,X3: int,B: int] :
      ( ( ord_less_eq_int @ A @ X3 )
     => ( ( ord_less_eq_int @ B @ X3 )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X3 ) ) ) ).

% le_supI
thf(fact_1979_sup__ge1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_1980_sup__ge1,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_1981_sup__ge1,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ X3 @ ( sup_sup_rat @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_1982_sup__ge1,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_1983_sup__ge1,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ X3 @ ( sup_sup_int @ X3 @ Y ) ) ).

% sup_ge1
thf(fact_1984_sup__ge2,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_1985_sup__ge2,axiom,
    ! [Y: set_nat,X3: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_1986_sup__ge2,axiom,
    ! [Y: rat,X3: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_1987_sup__ge2,axiom,
    ! [Y: nat,X3: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_1988_sup__ge2,axiom,
    ! [Y: int,X3: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X3 @ Y ) ) ).

% sup_ge2
thf(fact_1989_le__supI1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X3 @ A )
     => ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1990_le__supI1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ A )
     => ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1991_le__supI1,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X3 @ A )
     => ( ord_less_eq_rat @ X3 @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1992_le__supI1,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_1993_le__supI1,axiom,
    ! [X3: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X3 @ A )
     => ( ord_less_eq_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_1994_le__supI2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X3 @ B )
     => ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1995_le__supI2,axiom,
    ! [X3: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ B )
     => ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1996_le__supI2,axiom,
    ! [X3: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ X3 @ B )
     => ( ord_less_eq_rat @ X3 @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1997_le__supI2,axiom,
    ! [X3: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X3 @ B )
     => ( ord_less_eq_nat @ X3 @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_1998_le__supI2,axiom,
    ! [X3: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X3 @ B )
     => ( ord_less_eq_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_1999_sup_Omono,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A )
     => ( ( ord_le1268244103169919719at_nat @ D @ B )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ C @ D ) @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2000_sup_Omono,axiom,
    ! [C: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ( ord_less_eq_set_nat @ D @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2001_sup_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C @ D ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2002_sup_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2003_sup_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_2004_sup__mono,axiom,
    ! [A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,D: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ C )
     => ( ( ord_le1268244103169919719at_nat @ B @ D )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ A @ B ) @ ( sup_su5525570899277871387at_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2005_sup__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2006_sup__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2007_sup__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2008_sup__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C @ D ) ) ) ) ).

% sup_mono
thf(fact_2009_sup__least,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X3 )
     => ( ( ord_le1268244103169919719at_nat @ Z2 @ X3 )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_2010_sup__least,axiom,
    ! [Y: set_nat,X3: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( ord_less_eq_set_nat @ Z2 @ X3 )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_2011_sup__least,axiom,
    ! [Y: rat,X3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ( ord_less_eq_rat @ Z2 @ X3 )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_2012_sup__least,axiom,
    ! [Y: nat,X3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ Z2 @ X3 )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_2013_sup__least,axiom,
    ! [Y: int,X3: int,Z2: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ( ( ord_less_eq_int @ Z2 @ X3 )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z2 ) @ X3 ) ) ) ).

% sup_least
thf(fact_2014_le__iff__sup,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [X4: set_Pr4329608150637261639at_nat,Y3: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ X4 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2015_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( sup_sup_set_nat @ X4 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2016_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y3: rat] :
          ( ( sup_sup_rat @ X4 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2017_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( sup_sup_nat @ X4 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2018_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y3: int] :
          ( ( sup_sup_int @ X4 @ Y3 )
          = Y3 ) ) ) ).

% le_iff_sup
thf(fact_2019_sup_OorderE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( A
        = ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2020_sup_OorderE,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( A
        = ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2021_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2022_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2023_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_2024_sup_OorderI,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( A
        = ( sup_su5525570899277871387at_nat @ A @ B ) )
     => ( ord_le1268244103169919719at_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_2025_sup_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( sup_sup_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_2026_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_2027_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_2028_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_2029_sup__unique,axiom,
    ! [F: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_Pr4329608150637261639at_nat,Y4: set_Pr4329608150637261639at_nat,Z3: set_Pr4329608150637261639at_nat] :
              ( ( ord_le1268244103169919719at_nat @ Y4 @ X5 )
             => ( ( ord_le1268244103169919719at_nat @ Z3 @ X5 )
               => ( ord_le1268244103169919719at_nat @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
         => ( ( sup_su5525570899277871387at_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2030_sup__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: set_nat,Y4: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ Y4 @ X5 )
             => ( ( ord_less_eq_set_nat @ Z3 @ X5 )
               => ( ord_less_eq_set_nat @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
         => ( ( sup_sup_set_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2031_sup__unique,axiom,
    ! [F: rat > rat > rat,X3: rat,Y: rat] :
      ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: rat,Y4: rat,Z3: rat] :
              ( ( ord_less_eq_rat @ Y4 @ X5 )
             => ( ( ord_less_eq_rat @ Z3 @ X5 )
               => ( ord_less_eq_rat @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
         => ( ( sup_sup_rat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2032_sup__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y: nat] :
      ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ Y4 @ X5 )
             => ( ( ord_less_eq_nat @ Z3 @ X5 )
               => ( ord_less_eq_nat @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
         => ( ( sup_sup_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2033_sup__unique,axiom,
    ! [F: int > int > int,X3: int,Y: int] :
      ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ X5 @ ( F @ X5 @ Y4 ) )
     => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ Y4 @ ( F @ X5 @ Y4 ) )
       => ( ! [X5: int,Y4: int,Z3: int] :
              ( ( ord_less_eq_int @ Y4 @ X5 )
             => ( ( ord_less_eq_int @ Z3 @ X5 )
               => ( ord_less_eq_int @ ( F @ Y4 @ Z3 ) @ X5 ) ) )
         => ( ( sup_sup_int @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_2034_sup_Oabsorb1,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2035_sup_Oabsorb1,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( sup_sup_set_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2036_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2037_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2038_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_2039_sup_Oabsorb2,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A @ B )
     => ( ( sup_su5525570899277871387at_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2040_sup_Oabsorb2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( sup_sup_set_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2041_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2042_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2043_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_2044_sup__absorb1,axiom,
    ! [Y: set_Pr4329608150637261639at_nat,X3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ Y @ X3 )
     => ( ( sup_su5525570899277871387at_nat @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_2045_sup__absorb1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( sup_sup_set_nat @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_2046_sup__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ( sup_sup_rat @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_2047_sup__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( sup_sup_nat @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_2048_sup__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ( ( sup_sup_int @ X3 @ Y )
        = X3 ) ) ).

% sup_absorb1
thf(fact_2049_sup__absorb2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X3 @ Y )
     => ( ( sup_su5525570899277871387at_nat @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2050_sup__absorb2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( sup_sup_set_nat @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2051_sup__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( sup_sup_rat @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2052_sup__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( sup_sup_nat @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2053_sup__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( sup_sup_int @ X3 @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_2054_sup_OboundedE,axiom,
    ! [B: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A )
     => ~ ( ( ord_le1268244103169919719at_nat @ B @ A )
         => ~ ( ord_le1268244103169919719at_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_2055_sup_OboundedE,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_2056_sup_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_2057_sup_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_2058_sup_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% sup.boundedE
thf(fact_2059_sup_OboundedI,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,C: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ B @ A )
     => ( ( ord_le1268244103169919719at_nat @ C @ A )
       => ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2060_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2061_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2062_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2063_sup_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C ) @ A ) ) ) ).

% sup.boundedI
thf(fact_2064_sup_Oorder__iff,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B7: set_Pr4329608150637261639at_nat,A6: set_Pr4329608150637261639at_nat] :
          ( A6
          = ( sup_su5525570899277871387at_nat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_2065_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( A6
          = ( sup_sup_set_nat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_2066_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( A6
          = ( sup_sup_rat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_2067_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( A6
          = ( sup_sup_nat @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_2068_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( A6
          = ( sup_sup_int @ A6 @ B7 ) ) ) ) ).

% sup.order_iff
thf(fact_2069_sup_Ocobounded1,axiom,
    ! [A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ A @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2070_sup_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2071_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2072_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2073_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_2074_sup_Ocobounded2,axiom,
    ! [B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ B @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2075_sup_Ocobounded2,axiom,
    ! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2076_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2077_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2078_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_2079_sup_Oabsorb__iff1,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [B7: set_Pr4329608150637261639at_nat,A6: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_2080_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_2081_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( sup_sup_rat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_2082_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( sup_sup_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_2083_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( ( sup_sup_int @ A6 @ B7 )
          = A6 ) ) ) ).

% sup.absorb_iff1
thf(fact_2084_sup_Oabsorb__iff2,axiom,
    ( ord_le1268244103169919719at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat,B7: set_Pr4329608150637261639at_nat] :
          ( ( sup_su5525570899277871387at_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_2085_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( sup_sup_set_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_2086_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( sup_sup_rat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_2087_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( sup_sup_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_2088_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] :
          ( ( sup_sup_int @ A6 @ B7 )
          = B7 ) ) ) ).

% sup.absorb_iff2
thf(fact_2089_sup_OcoboundedI1,axiom,
    ! [C: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ A )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2090_sup_OcoboundedI1,axiom,
    ! [C: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ A )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2091_sup_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2092_sup_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2093_sup_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_2094_sup_OcoboundedI2,axiom,
    ! [C: set_Pr4329608150637261639at_nat,B: set_Pr4329608150637261639at_nat,A: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C @ B )
     => ( ord_le1268244103169919719at_nat @ C @ ( sup_su5525570899277871387at_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2095_sup_OcoboundedI2,axiom,
    ! [C: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C @ B )
     => ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2096_sup_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2097_sup_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2098_sup_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_2099_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2 = N )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ 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 @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_2100_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
         => ( vEBT_invar_vebt @ X5 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
         => ( ( M2
              = ( suc @ N ) )
           => ( ( Deg
                = ( plus_plus_nat @ N @ M2 ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ 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 @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_2101_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N3: nat,TreeList4: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N3 ) ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) ) ) ).

% in_children_def
thf(fact_2102_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X3: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X3 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_2103_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ 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 @ TreeList ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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_2104_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList: 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 @ TreeList @ Summary ) @ X3 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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_2105_pred__list__to__short,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( 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 @ TreeList @ Summary ) @ X3 )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_2106_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X3 )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList ) @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_2107_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ Summary ) @ X3 ) ) ) ) ).

% both_member_options_ding
thf(fact_2108_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_2109_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_2110_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_2111_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_2112_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_2113_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_2114_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_2115_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_2116_low__inv,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
        = X3 ) ) ).

% low_inv
thf(fact_2117_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 ) ) ).

% Suc_diff_diff
thf(fact_2118_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_2119_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_2120_diff__diff__left,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% diff_diff_left
thf(fact_2121_power__minus__is__div,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ A @ B ) )
        = ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% power_minus_is_div
thf(fact_2122_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_2123_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X4: nat,N3: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% high_def
thf(fact_2124_high__inv,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X3 ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_2125_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_2126_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_2127_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_2128_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_2129_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_2130_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2131_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2132_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Z2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2133_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2134_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W: num,Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Z2 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2135_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_2136_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_2137_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_2138_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2139_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2140_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_2141_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_2142_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_2143_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2144_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2145_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H2: nat,L2: nat,D3: nat] : ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D3 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_2146_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_2147_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_2148_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_2149_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_2150_nat__1__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2151_nat__mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = one_one_nat )
      = ( ( M2 = one_one_nat )
        & ( N = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2152_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V2: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_2153_distrib__right__numeral,axiom,
    ! [A: real,B: real,V2: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V2 ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_2154_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V2: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_2155_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V2: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V2 ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_2156_distrib__right__numeral,axiom,
    ! [A: int,B: int,V2: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_2157_distrib__left__numeral,axiom,
    ! [V2: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2158_distrib__left__numeral,axiom,
    ! [V2: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2159_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2160_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2161_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2162_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V2: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2163_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V2: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V2 ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2164_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V2: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2165_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V2: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V2 ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_2166_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2167_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2168_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2169_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_2170_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) @ I )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_2171_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K2 ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_2172_mult__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ M2 @ ( suc @ N ) )
      = ( plus_plus_nat @ M2 @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc_right
thf(fact_2173_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [M: nat] :
              ( ( ( some_nat @ M )
                = ( vEBT_vebt_mint @ Summary ) )
              & ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% mintlistlength
thf(fact_2174_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2175_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2176_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2177_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2178_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2179_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2180_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2181_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2182_power__add__numeral2,axiom,
    ! [A: complex,M2: num,N: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2183_power__add__numeral2,axiom,
    ! [A: real,M2: num,N: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2184_power__add__numeral2,axiom,
    ! [A: rat,M2: num,N: num,B: rat] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2185_power__add__numeral2,axiom,
    ! [A: nat,M2: num,N: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2186_power__add__numeral2,axiom,
    ! [A: int,M2: num,N: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2187_power__add__numeral,axiom,
    ! [A: complex,M2: num,N: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_2188_power__add__numeral,axiom,
    ! [A: real,M2: num,N: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_2189_power__add__numeral,axiom,
    ! [A: rat,M2: num,N: num] :
      ( ( times_times_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_rat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_2190_power__add__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_2191_power__add__numeral,axiom,
    ! [A: int,M2: num,N: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% power_add_numeral
thf(fact_2192_diff__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
      = ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% diff_mult_distrib2
thf(fact_2193_diff__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M2 @ N ) @ K2 )
      = ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% diff_mult_distrib
thf(fact_2194_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2195_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2196_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2197_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2198_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2199_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2200_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2201_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2202_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A6: real,B7: real] : ( times_times_real @ B7 @ A6 ) ) ) ).

% mult.commute
thf(fact_2203_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A6: rat,B7: rat] : ( times_times_rat @ B7 @ A6 ) ) ) ).

% mult.commute
thf(fact_2204_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A6: nat,B7: nat] : ( times_times_nat @ B7 @ A6 ) ) ) ).

% mult.commute
thf(fact_2205_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A6: int,B7: int] : ( times_times_int @ B7 @ A6 ) ) ) ).

% mult.commute
thf(fact_2206_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2207_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2208_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2209_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2210_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2211_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2212_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2213_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).

% combine_common_factor
thf(fact_2214_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_2215_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2216_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2217_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_2218_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2219_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2220_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2221_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2222_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2223_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2224_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2225_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2226_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2227_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2228_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2229_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2230_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2231_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2232_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_2233_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_2234_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_2235_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( 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 ) @ M2 ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_2236_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_2237_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_2238_power__divide,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_divide
thf(fact_2239_power__divide,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N )
      = ( divide_divide_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_divide
thf(fact_2240_power__divide,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ A @ B ) @ N )
      = ( divide_divide_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_divide
thf(fact_2241_zero__induct__lemma,axiom,
    ! [P: nat > $o,K2: nat,I: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ ( minus_minus_nat @ K2 @ I ) ) ) ) ).

% zero_induct_lemma
thf(fact_2242_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_2243_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_2244_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_2245_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_2246_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_2247_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2248_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2249_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_2250_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2251_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2252_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_2253_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ J @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_2254_power__commuting__commutes,axiom,
    ! [X3: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X3 @ Y )
        = ( times_times_complex @ Y @ X3 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2255_power__commuting__commutes,axiom,
    ! [X3: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X3 @ Y )
        = ( times_times_real @ Y @ X3 ) )
     => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2256_power__commuting__commutes,axiom,
    ! [X3: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X3 @ Y )
        = ( times_times_rat @ Y @ X3 ) )
     => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2257_power__commuting__commutes,axiom,
    ! [X3: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X3 @ Y )
        = ( times_times_nat @ Y @ X3 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2258_power__commuting__commutes,axiom,
    ! [X3: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X3 @ Y )
        = ( times_times_int @ Y @ X3 ) )
     => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X3 @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2259_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N )
      = ( times_times_complex @ ( power_power_complex @ A @ N ) @ ( power_power_complex @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2260_power__mult__distrib,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N )
      = ( times_times_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2261_power__mult__distrib,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( times_times_rat @ A @ B ) @ N )
      = ( times_times_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2262_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N )
      = ( times_times_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2263_power__mult__distrib,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N )
      = ( times_times_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ).

% power_mult_distrib
thf(fact_2264_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_2265_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_2266_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_2267_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_2268_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_2269_Suc__mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
        = ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( M2 = N ) ) ).

% Suc_mult_cancel1
thf(fact_2270_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_2271_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_2272_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_2273_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_2274_Nat_Odiff__diff__eq,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_2275_le__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% le_diff_iff
thf(fact_2276_eq__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ( minus_minus_nat @ M2 @ K2 )
            = ( minus_minus_nat @ N @ K2 ) )
          = ( M2 = N ) ) ) ) ).

% eq_diff_iff
thf(fact_2277_power__mult,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_2278_power__mult,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_real @ ( power_power_real @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_2279_power__mult,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_int @ ( power_power_int @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_2280_power__mult,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M2 @ N ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M2 ) @ N ) ) ).

% power_mult
thf(fact_2281_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_2282_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_2283_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_2284_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_2285_mult__le__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ).

% mult_le_mono2
thf(fact_2286_mult__le__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ).

% mult_le_mono1
thf(fact_2287_mult__le__mono,axiom,
    ! [I: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2288_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_2289_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_2290_left__add__mult__distrib,axiom,
    ! [I: nat,U: nat,J: nat,K2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K2 ) ) ).

% left_add_mult_distrib
thf(fact_2291_add__mult__distrib2,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% add_mult_distrib2
thf(fact_2292_add__mult__distrib,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M2 @ N ) @ K2 )
      = ( plus_plus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) ) ) ).

% add_mult_distrib
thf(fact_2293_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_2294_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_2295_eq__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2296_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2297_eq__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_2298_eq__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2299_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2300_eq__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2301_square__diff__square__factored,axiom,
    ! [X3: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_real @ X3 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2302_square__diff__square__factored,axiom,
    ! [X3: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( minus_minus_rat @ X3 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2303_square__diff__square__factored,axiom,
    ! [X3: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X3 @ Y ) @ ( minus_minus_int @ X3 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2304_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_2305_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_2306_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2307_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2308_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2309_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2310_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2311_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2312_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_2313_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_2314_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_2315_less__add__iff2,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2316_less__add__iff2,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2317_less__add__iff2,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2318_less__add__iff1,axiom,
    ! [A: real,E: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2319_less__add__iff1,axiom,
    ! [A: rat,E: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2320_less__add__iff1,axiom,
    ! [A: int,E: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_2329_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_2330_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_2331_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_2332_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_2333_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_2334_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_2335_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_2336_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_2337_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_2338_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_2339_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_2340_left__right__inverse__power,axiom,
    ! [X3: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X3 @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_2341_left__right__inverse__power,axiom,
    ! [X3: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X3 @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_2342_left__right__inverse__power,axiom,
    ! [X3: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X3 @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2343_left__right__inverse__power,axiom,
    ! [X3: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X3 @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2344_left__right__inverse__power,axiom,
    ! [X3: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X3 @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2345_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_2346_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_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_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_2356_less__diff__iff,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N @ K2 ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_2357_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_2358_less__diff__conv,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ).

% less_diff_conv
thf(fact_2359_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_2360_Suc__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_mult_less_cancel1
thf(fact_2361_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K2 )
        = ( J
          = ( plus_plus_nat @ K2 @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_2362_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K2 ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_2363_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_2364_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_2365_le__diff__conv,axiom,
    ! [J: nat,K2: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ).

% le_diff_conv
thf(fact_2366_power__add,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).

% power_add
thf(fact_2367_power__add,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).

% power_add
thf(fact_2368_power__add,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ).

% power_add
thf(fact_2369_power__add,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% power_add
thf(fact_2370_power__add,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M2 @ N ) )
      = ( times_times_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).

% power_add
thf(fact_2371_Suc__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_mult_le_cancel1
thf(fact_2372_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_2373_mult__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ).

% mult_Suc
thf(fact_2374_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_2375_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_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_less__diff__conv2,axiom,
    ! [K2: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K2 @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K2 ) @ I )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_2388_left__add__twice,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2389_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2390_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2391_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2392_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_2393_mult__2__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_2394_mult__2__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_2395_mult__2__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_2396_mult__2__right,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_2397_mult__2__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_2398_mult__2,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_2399_mult__2,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_2400_mult__2,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_2401_mult__2,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_2402_mult__2,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_diff__le__diff__pow,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ ( minus_minus_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) ) ) ) ).

% diff_le_diff_pow
thf(fact_2418_power2__sum,axiom,
    ! [X3: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_sum
thf(fact_2419_power2__sum,axiom,
    ! [X3: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_sum
thf(fact_2420_power2__sum,axiom,
    ! [X3: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_sum
thf(fact_2421_power2__sum,axiom,
    ! [X3: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_sum
thf(fact_2422_power2__sum,axiom,
    ! [X3: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_sum
thf(fact_2423_power2__diff,axiom,
    ! [X3: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_diff
thf(fact_2424_power2__diff,axiom,
    ! [X3: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_diff
thf(fact_2425_power2__diff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_diff
thf(fact_2426_power2__diff,axiom,
    ! [X3: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X3 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y ) ) ) ).

% power2_diff
thf(fact_2427_add__le__add__imp__diff__le,axiom,
    ! [I: real,K2: real,N: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J @ K2 ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2428_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K2: rat,N: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J @ K2 ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2429_add__le__add__imp__diff__le,axiom,
    ! [I: nat,K2: nat,N: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2430_add__le__add__imp__diff__le,axiom,
    ! [I: int,K2: int,N: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K2 ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2431_add__le__imp__le__diff,axiom,
    ! [I: real,K2: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K2 ) @ N )
     => ( ord_less_eq_real @ I @ ( minus_minus_real @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2432_add__le__imp__le__diff,axiom,
    ! [I: rat,K2: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K2 ) @ N )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2433_add__le__imp__le__diff,axiom,
    ! [I: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K2 ) @ N )
     => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2434_add__le__imp__le__diff,axiom,
    ! [I: int,K2: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K2 ) @ N )
     => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2435_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_2436_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_2437_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_2438_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_2439_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2440_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2441_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2442_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2443_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_2444_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_2445_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_2446_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_2447_add__self__div__2,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M2 ) ).

% add_self_div_2
thf(fact_2448_div2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_2449_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: 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 @ TreeList @ 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 @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ).

% nested_mint
thf(fact_2450_sum__squares__bound,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_2451_sum__squares__bound,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_2452_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_2453_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_2454_div__exp__eq,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% div_exp_eq
thf(fact_2455_div__exp__eq,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% div_exp_eq
thf(fact_2456_field__less__half__sum,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ X3 @ Y )
     => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2457_field__less__half__sum,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ X3 @ Y )
     => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2458_div__nat__eqI,axiom,
    ! [N: nat,Q3: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q3 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M2 @ N )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_2459_Suc__double__not__eq__double,axiom,
    ! [M2: nat,N: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% Suc_double_not_eq_double
thf(fact_2460_mul__shift,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ( times_times_nat @ X3 @ Y )
        = Z2 )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z2 ) ) ) ).

% mul_shift
thf(fact_2461_semiring__norm_I13_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M2 @ N ) ) ) ) ).

% semiring_norm(13)
thf(fact_2462_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_2463_semiring__norm_I11_J,axiom,
    ! [M2: num] :
      ( ( times_times_num @ M2 @ one )
      = M2 ) ).

% semiring_norm(11)
thf(fact_2464_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2465_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2466_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_2467_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_2468_power__mult__numeral,axiom,
    ! [A: nat,M2: num,N: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_2469_power__mult__numeral,axiom,
    ! [A: real,M2: num,N: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_2470_power__mult__numeral,axiom,
    ! [A: int,M2: num,N: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_2471_power__mult__numeral,axiom,
    ! [A: complex,M2: num,N: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% power_mult_numeral
thf(fact_2472_diff__commute,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K2 ) @ J ) ) ).

% diff_commute
thf(fact_2473_div__mult2__numeral__eq,axiom,
    ! [A: nat,K2: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K2 @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_2474_div__mult2__numeral__eq,axiom,
    ! [A: int,K2: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K2 @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_2475_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_2476_option_Osel,axiom,
    ! [X2: nat] :
      ( ( the_nat @ ( some_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_2477_option_Osel,axiom,
    ! [X2: num] :
      ( ( the_num @ ( some_num @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_2478_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_2479_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_2480_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_2481_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_2482_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_2483_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2484_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2485_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_2486_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_2487_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2488_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2489_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_2490_div__le__mono,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ K2 ) @ ( divide_divide_nat @ N @ K2 ) ) ) ).

% div_le_mono
thf(fact_2491_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).

% div_le_dividend
thf(fact_2492_mult__diff__mult,axiom,
    ! [X3: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2493_mult__diff__mult,axiom,
    ! [X3: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2494_mult__diff__mult,axiom,
    ! [X3: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2495_Suc__div__le__mono,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ ( divide_divide_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_div_le_mono
thf(fact_2496_times__div__less__eq__dividend,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) @ M2 ) ).

% times_div_less_eq_dividend
thf(fact_2497_div__times__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) @ M2 ) ).

% div_times_less_eq_dividend
thf(fact_2498_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_2499_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_2500_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_2501_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_2502_double__not__eq__Suc__double,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% double_not_eq_Suc_double
thf(fact_2503_del__x__mi__lets__in__not__minNull,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( ord_less_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_not_minNull
thf(fact_2504_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X3 )
        & ( ord_less_eq_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X3 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_newnode_not_nil
thf(fact_2505_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_2506_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_2507_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_2508_vebt__succ_Osimps_I3_J,axiom,
    ! [Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,Va: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz ) @ Va )
      = none_nat ) ).

% vebt_succ.simps(3)
thf(fact_2509_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_2510_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_2511_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_2512_pred__lesseq__max,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,Mi: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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_2513_pred__less__length__list,axiom,
    ! [Deg: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X3 @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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_2514_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,TreeList: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X3 )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_2515_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_2516_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_2517_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X3 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y3: nat] :
                ( ( vEBT_vebt_member @ T @ Y3 )
                & ( ord_less_nat @ X3 @ Y3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_2518_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X3 )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y3: nat] :
                ( ( vEBT_vebt_member @ T @ Y3 )
                & ( ord_less_nat @ Y3 @ X3 ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_2519_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y5: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : ( Y5 = Z )
          @ A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_2520_singleton__conv2,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ( ^ [Y5: complex,Z: complex] : ( Y5 = Z )
          @ A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv2
thf(fact_2521_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y5: set_nat,Z: set_nat] : ( Y5 = Z )
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_2522_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z )
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_2523_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y5: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y5 = Z )
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_2524_singleton__conv2,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ( ^ [Y5: $o,Z: $o] : ( Y5 = Z )
          @ A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv2
thf(fact_2525_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y5: nat,Z: nat] : ( Y5 = Z )
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_2526_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y5: int,Z: int] : ( Y5 = Z )
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_2527_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] : ( X4 = A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_2528_singleton__conv,axiom,
    ! [A: complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( X4 = A ) )
      = ( insert_complex @ A @ bot_bot_set_complex ) ) ).

% singleton_conv
thf(fact_2529_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] : ( X4 = A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_2530_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] : ( X4 = A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_2531_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( X4 = A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_2532_singleton__conv,axiom,
    ! [A: $o] :
      ( ( collect_o
        @ ^ [X4: $o] : ( X4 = A ) )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% singleton_conv
thf(fact_2533_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( X4 = A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_2534_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X4: int] : ( X4 = A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_2535_del__x__not__mia,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X3 )
        & ( ord_less_eq_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X3 = Ma )
                            @ ( if_nat
                              @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                = none_nat )
                              @ Mi
                              @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X3 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_2536_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X3 )
        & ( ord_less_eq_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X3 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_2537_del__x__not__mi,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X3 )
        & ( ord_less_eq_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X3 = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
                      & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X3 = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi
thf(fact_2538_del__in__range,axiom,
    ! [Mi: nat,X3: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X3 )
        & ( ord_less_eq_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X3 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X3 != Mi )
                           => ( X3 = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X3 = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X3 != Mi )
                           => ( X3 = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_2539_del__x__mi,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat] :
      ( ( ( X3 = Mi )
        & ( ord_less_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                      @ ( vEBT_Node
                        @ ( some_P7363390416028606310at_nat
                          @ ( product_Pair_nat_nat @ Xn
                            @ ( if_nat @ ( Xn = Ma )
                              @ ( if_nat
                                @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                  = none_nat )
                                @ Xn
                                @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                        @ ( vEBT_vebt_delete @ Summary @ H ) )
                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_2540_del__x__mi__lets__in,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( ord_less_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ ( vEBT_vebt_delete @ Summary @ H ) ) ) )
                        & ( ~ ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xn @ ( if_nat @ ( Xn = Ma ) @ ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ H ) ) ) ) @ Ma ) ) ) @ Deg @ Newlist @ Summary ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in
thf(fact_2541_del__x__mi__lets__in__minNull,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( ord_less_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( Xn
                = ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) )
             => ( ( ( vEBT_VEBT_low @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                  = L )
               => ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList @ H @ Newnode ) )
                     => ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( Sn
                            = ( vEBT_vebt_delete @ Summary @ H ) )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_2542_del__x__mia,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X3 = Mi )
        & ( ord_less_nat @ X3 @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
              @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                      @ ( if_nat
                        @ ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                          = Ma )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_2543_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X3: nat,Ma: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X3 )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X3 )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_2544_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 )
            @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2545_minus__set__def,axiom,
    ( minus_minus_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ( minus_minus_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A5 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2546_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 )
            @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2547_minus__set__def,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( minus_8727706125548526216plex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
            @ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2548_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2549_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2550_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2551_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_2552_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A5 )
              & ~ ( member_real @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2553_set__diff__eq,axiom,
    ( minus_minus_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A5 )
              & ~ ( member_o @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2554_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A5 )
              & ~ ( member_int @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2555_set__diff__eq,axiom,
    ( minus_811609699411566653omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A5 )
              & ~ ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2556_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A5 )
              & ~ ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2557_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A5 )
              & ~ ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2558_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A5 )
              & ~ ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2559_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              & ~ ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_2560_real__arch__pow,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ? [N2: nat] : ( ord_less_real @ Y @ ( power_power_real @ X3 @ N2 ) ) ) ).

% real_arch_pow
thf(fact_2561_sup__Un__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( sup_sup_num_num_o
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ ( sup_su4061117120043295689um_num @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_2562_sup__Un__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( sup_sup_nat_num_o
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ ( sup_su2042722026077122175at_num @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_2563_sup__Un__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( sup_su6327502436637775413at_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_2564_sup__Un__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( sup_sup_int_int_o
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R )
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ ( sup_su6024340866399070445nt_int @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_2565_sup__Un__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su7519161239522478338_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ ( sup_su5525570899277871387at_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_2566_sup__Un__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ( sup_sup_complex_o
        @ ^ [X4: complex] : ( member_complex @ X4 @ R )
        @ ^ [X4: complex] : ( member_complex @ X4 @ S3 ) )
      = ( ^ [X4: complex] : ( member_complex @ X4 @ ( sup_sup_set_complex @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2567_sup__Un__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( sup_sup_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R )
        @ ^ [X4: real] : ( member_real @ X4 @ S3 ) )
      = ( ^ [X4: real] : ( member_real @ X4 @ ( sup_sup_set_real @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2568_sup__Un__eq,axiom,
    ! [R: set_o,S3: set_o] :
      ( ( sup_sup_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S3 ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( sup_sup_set_o @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2569_sup__Un__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( sup_sup_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S3 ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( sup_sup_set_int @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2570_sup__Un__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( sup_sup_set_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2571_sup__Un__eq,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su2080679670758317954_nat_o
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ R )
        @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ S3 ) )
      = ( ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ ( sup_su5525570899277871387at_nat @ R @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_2572_Un__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A5 )
              | ( member_real @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2573_Un__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A5 )
              | ( member_o @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2574_Un__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A5 )
              | ( member_int @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2575_Un__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A5 )
              | ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2576_Un__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A5 )
              | ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2577_Un__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A5 )
              | ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2578_Un__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A5 )
              | ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2579_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              | ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2580_Un__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( member8757157785044589968at_nat @ X4 @ A5 )
              | ( member8757157785044589968at_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_2581_sup__set__def,axiom,
    ( sup_sup_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( sup_sup_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 )
            @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2582_sup__set__def,axiom,
    ( sup_sup_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ( sup_sup_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A5 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2583_sup__set__def,axiom,
    ( sup_sup_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( sup_sup_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 )
            @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2584_sup__set__def,axiom,
    ( sup_sup_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( sup_sup_complex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
            @ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2585_sup__set__def,axiom,
    ( sup_su6327502436637775413at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( sup_su798857527126471912_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2586_sup__set__def,axiom,
    ( sup_sup_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( sup_sup_set_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2587_sup__set__def,axiom,
    ( sup_sup_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( sup_sup_list_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2588_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2589_sup__set__def,axiom,
    ( sup_su5525570899277871387at_nat
    = ( ^ [A5: set_Pr4329608150637261639at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ( sup_su2080679670758317954_nat_o
            @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ A5 )
            @ ^ [X4: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_2590_Collect__disj__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X4: complex] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_complex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2591_Collect__disj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2592_Collect__disj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2593_Collect__disj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2594_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2595_Collect__disj__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( collec6321179662152712658at_nat @ P ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_2596_pred__subset__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ( ord_le4573692005234683329plex_o
        @ ^ [X4: complex] : ( member_complex @ X4 @ R )
        @ ^ [X4: complex] : ( member_complex @ X4 @ S3 ) )
      = ( ord_le211207098394363844omplex @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_2597_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R )
        @ ^ [X4: real] : ( member_real @ X4 @ S3 ) )
      = ( ord_less_eq_set_real @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_2598_pred__subset__eq,axiom,
    ! [R: set_o,S3: set_o] :
      ( ( ord_less_eq_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S3 ) )
      = ( ord_less_eq_set_o @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_2599_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S3 ) )
      = ( ord_less_eq_set_int @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_2600_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
      = ( ord_less_eq_set_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_2601_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
          @ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_2602_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A5 )
          @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_2603_less__eq__set__def,axiom,
    ( ord_less_eq_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( ord_less_eq_o_o
          @ ^ [X4: $o] : ( member_o @ X4 @ A5 )
          @ ^ [X4: $o] : ( member_o @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_2604_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A5 )
          @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_2605_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_2606_Collect__subset,axiom,
    ! [A4: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X4: real] :
            ( ( member_real @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2607_Collect__subset,axiom,
    ! [A4: set_o,P: $o > $o] :
      ( ord_less_eq_set_o
      @ ( collect_o
        @ ^ [X4: $o] :
            ( ( member_o @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2608_Collect__subset,axiom,
    ! [A4: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X4: int] :
            ( ( member_int @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2609_Collect__subset,axiom,
    ! [A4: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X4: complex] :
            ( ( member_complex @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2610_Collect__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2611_Collect__subset,axiom,
    ! [A4: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2612_Collect__subset,axiom,
    ! [A4: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( member_list_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2613_Collect__subset,axiom,
    ! [A4: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( P @ X4 ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_2614_bot__empty__eq2,axiom,
    ( bot_bo394778441745866138_nat_o
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ bot_bo228742789529271731at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_2615_bot__empty__eq2,axiom,
    ( bot_bot_num_num_o
    = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ bot_bo9056780473022590049um_num ) ) ) ).

% bot_empty_eq2
thf(fact_2616_bot__empty__eq2,axiom,
    ( bot_bot_nat_num_o
    = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ bot_bo7038385379056416535at_num ) ) ) ).

% bot_empty_eq2
thf(fact_2617_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_2618_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_2619_pred__equals__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_2620_pred__equals__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_2621_pred__equals__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_2622_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_2623_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R ) )
        = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_2624_empty__def,axiom,
    ( bot_bot_set_complex
    = ( collect_complex
      @ ^ [X4: complex] : $false ) ) ).

% empty_def
thf(fact_2625_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X4: set_nat] : $false ) ) ).

% empty_def
thf(fact_2626_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X4: list_nat] : $false ) ) ).

% empty_def
thf(fact_2627_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X4: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_2628_empty__def,axiom,
    ( bot_bot_set_o
    = ( collect_o
      @ ^ [X4: $o] : $false ) ) ).

% empty_def
thf(fact_2629_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X4: nat] : $false ) ) ).

% empty_def
thf(fact_2630_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X4: int] : $false ) ) ).

% empty_def
thf(fact_2631_insert__Collect,axiom,
    ! [A: int,P: int > $o] :
      ( ( insert_int @ A @ ( collect_int @ P ) )
      = ( collect_int
        @ ^ [U2: int] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2632_insert__Collect,axiom,
    ! [A: $o,P: $o > $o] :
      ( ( insert_o @ A @ ( collect_o @ P ) )
      = ( collect_o
        @ ^ [U2: $o] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2633_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( collec6321179662152712658at_nat
        @ ^ [U2: produc3843707927480180839at_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2634_insert__Collect,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( insert_nat @ A @ ( collect_nat @ P ) )
      = ( collect_nat
        @ ^ [U2: nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2635_insert__Collect,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( insert_complex @ A @ ( collect_complex @ P ) )
      = ( collect_complex
        @ ^ [U2: complex] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2636_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2637_insert__Collect,axiom,
    ! [A: set_nat,P: set_nat > $o] :
      ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
      = ( collect_set_nat
        @ ^ [U2: set_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2638_insert__Collect,axiom,
    ! [A: list_nat,P: list_nat > $o] :
      ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
      = ( collect_list_nat
        @ ^ [U2: list_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_2639_insert__compr,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A6: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X4: produc3843707927480180839at_nat] :
              ( ( X4 = A6 )
              | ( member8757157785044589968at_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2640_insert__compr,axiom,
    ( insert_real
    = ( ^ [A6: real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( X4 = A6 )
              | ( member_real @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2641_insert__compr,axiom,
    ( insert_o
    = ( ^ [A6: $o,B5: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( X4 = A6 )
              | ( member_o @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2642_insert__compr,axiom,
    ( insert_int
    = ( ^ [A6: int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( X4 = A6 )
              | ( member_int @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2643_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A6: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( X4 = A6 )
              | ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2644_insert__compr,axiom,
    ( insert_complex
    = ( ^ [A6: complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( X4 = A6 )
              | ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2645_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A6: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( X4 = A6 )
              | ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2646_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A6: set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( X4 = A6 )
              | ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2647_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A6: list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( X4 = A6 )
              | ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_2648_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( ord_less_complex_o
          @ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
          @ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_2649_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_real_o
          @ ^ [X4: real] : ( member_real @ X4 @ A5 )
          @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_2650_less__set__def,axiom,
    ( ord_less_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( ord_less_o_o
          @ ^ [X4: $o] : ( member_o @ X4 @ A5 )
          @ ^ [X4: $o] : ( member_o @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_2651_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
          @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_2652_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_int_o
          @ ^ [X4: int] : ( member_int @ X4 @ A5 )
          @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_2653_Collect__conv__if2,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_2654_Collect__conv__if2,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if2
thf(fact_2655_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_2656_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_2657_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_2658_Collect__conv__if2,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if2
thf(fact_2659_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_2660_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( A = X4 )
                & ( P @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_2661_Collect__conv__if,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_2662_Collect__conv__if,axiom,
    ! [P: complex > $o,A: complex] :
      ( ( ( P @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_complex @ A @ bot_bot_set_complex ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_complex
            @ ^ [X4: complex] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_complex ) ) ) ).

% Collect_conv_if
thf(fact_2663_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X4: set_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_2664_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X4: list_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_2665_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_2666_Collect__conv__if,axiom,
    ! [P: $o > $o,A: $o] :
      ( ( ( P @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_o @ A @ bot_bot_set_o ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_o
            @ ^ [X4: $o] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_o ) ) ) ).

% Collect_conv_if
thf(fact_2667_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X4: nat] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_2668_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X4: int] :
                ( ( X4 = A )
                & ( P @ X4 ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_2669_insert__def,axiom,
    ( insert_int
    = ( ^ [A6: int] :
          ( sup_sup_set_int
          @ ( collect_int
            @ ^ [X4: int] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2670_insert__def,axiom,
    ( insert_o
    = ( ^ [A6: $o] :
          ( sup_sup_set_o
          @ ( collect_o
            @ ^ [X4: $o] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2671_insert__def,axiom,
    ( insert_complex
    = ( ^ [A6: complex] :
          ( sup_sup_set_complex
          @ ( collect_complex
            @ ^ [X4: complex] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2672_insert__def,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A6: product_prod_nat_nat] :
          ( sup_su6327502436637775413at_nat
          @ ( collec3392354462482085612at_nat
            @ ^ [X4: product_prod_nat_nat] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2673_insert__def,axiom,
    ( insert_set_nat
    = ( ^ [A6: set_nat] :
          ( sup_sup_set_set_nat
          @ ( collect_set_nat
            @ ^ [X4: set_nat] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2674_insert__def,axiom,
    ( insert_list_nat
    = ( ^ [A6: list_nat] :
          ( sup_sup_set_list_nat
          @ ( collect_list_nat
            @ ^ [X4: list_nat] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2675_insert__def,axiom,
    ( insert_nat
    = ( ^ [A6: nat] :
          ( sup_sup_set_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2676_insert__def,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A6: produc3843707927480180839at_nat] :
          ( sup_su5525570899277871387at_nat
          @ ( collec6321179662152712658at_nat
            @ ^ [X4: produc3843707927480180839at_nat] : ( X4 = A6 ) ) ) ) ) ).

% insert_def
thf(fact_2677_pred__subset__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( ord_le3935385432712749774_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le1268244103169919719at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_2678_pred__subset__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( ord_le6124364862034508274_num_o
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le880128212290418581um_num @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_2679_pred__subset__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( ord_le3404735783095501756_num_o
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le8085105155179020875at_num @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_2680_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_2681_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R )
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_2682_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_2683_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_2684_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_2685_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_2686_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_2687_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_2688_power__numeral__even,axiom,
    ! [Z2: complex,W: num] :
      ( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2689_power__numeral__even,axiom,
    ! [Z2: real,W: num] :
      ( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2690_power__numeral__even,axiom,
    ! [Z2: rat,W: num] :
      ( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2691_power__numeral__even,axiom,
    ! [Z2: nat,W: num] :
      ( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2692_power__numeral__even,axiom,
    ! [Z2: int,W: num] :
      ( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_2693_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V2: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V2 ) @ TreeList @ S ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_2694_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList @ Vd ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_2695_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc ) @ X3 )
      = ( ( X3 = Mi )
        | ( X3 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_2696_vebt__succ_Osimps_I6_J,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X3 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X3 @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ 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 @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ ( nth_VEBT_VEBT @ TreeList @ ( 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_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_2697_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X3: nat,Mi: nat,Va: nat,TreeList: 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 ) ) @ TreeList @ 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 ) ) @ TreeList @ 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 @ TreeList ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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_2698_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X4: nat,Y3: nat] :
          ( ( member_nat @ Y3 @ Xs )
          & ( ord_less_nat @ X4 @ Y3 )
          & ! [Z4: nat] :
              ( ( member_nat @ Z4 @ Xs )
             => ( ( ord_less_nat @ X4 @ Z4 )
               => ( ord_less_eq_nat @ Y3 @ Z4 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_2699_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X4: nat,Y3: nat] :
          ( ( member_nat @ Y3 @ Xs )
          & ( ord_less_nat @ Y3 @ X4 )
          & ! [Z4: nat] :
              ( ( member_nat @ Z4 @ Xs )
             => ( ( ord_less_nat @ Z4 @ X4 )
               => ( ord_less_eq_nat @ Z4 @ Y3 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_2700_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A6 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_2701_discrete,axiom,
    ( ord_less_int
    = ( ^ [A6: int] : ( ord_less_eq_int @ ( plus_plus_int @ A6 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_2702_vebt__delete_Osimps_I7_J,axiom,
    ! [X3: nat,Mi: nat,Ma: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X3 @ Mi )
          | ( ord_less_nat @ Ma @ X3 ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X3 @ Mi )
            | ( ord_less_nat @ Ma @ X3 ) )
       => ( ( ( ( X3 = Mi )
              & ( X3 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) )
          & ( ~ ( ( X3 = Mi )
                & ( X3 = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X3 )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X3 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X3 != Mi )
                             => ( X3 = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X3 = Mi ) @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X3 = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X3 != Mi )
                             => ( X3 = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X3 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X3 = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ 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 ) ) @ TreeList @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_2703_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: 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 ) ) @ TreeList @ 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 @ TreeList ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_2704_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_2705_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_2706_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_2707_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_2708_insert__simp__norm,axiom,
    ! [X3: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( 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 @ TreeList @ Summary ) @ X3 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X3 @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList @ ( 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_2709_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList: 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 @ TreeList ) )
     => ( ( 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 @ TreeList @ Summary ) @ X3 )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X3 @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_2710_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_2711_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_2712_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_2713_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_2714_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_2715_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_2716_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_2717_list__update__beyond,axiom,
    ! [Xs2: list_int,I: nat,X3: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I )
     => ( ( list_update_int @ Xs2 @ I @ X3 )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_2718_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_2719_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_2720_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_2721_length__list__update,axiom,
    ! [Xs2: list_int,I: nat,X3: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I @ X3 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_2722_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
           => ~ ( ( Xa2 = Mi2 )
                | ( Xa2 = Ma2 ) ) )
       => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
             => ~ ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
         => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [Vd2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_2723_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_2724_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_2725_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_2726_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_2727_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_2728_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_2729_add_Oright__neutral,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ A @ zero_zero_literal )
      = A ) ).

% add.right_neutral
thf(fact_2730_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_2731_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_2732_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_2733_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_2734_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_2735_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_2736_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_2737_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_2738_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_2739_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_2740_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_2741_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_2742_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_2743_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_2744_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_2745_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_2746_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_2747_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_2748_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_2749_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_2750_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_2751_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_2752_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_2753_add__eq__0__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y )
        = zero_zero_nat )
      = ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_2754_zero__eq__add__iff__both__eq__0,axiom,
    ! [X3: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X3 @ Y ) )
      = ( ( X3 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_2755_add__0,axiom,
    ! [A: literal] :
      ( ( plus_plus_literal @ zero_zero_literal @ A )
      = A ) ).

% add_0
thf(fact_2756_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_2757_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_2758_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_2759_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_2760_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_2761_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_2762_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_2763_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_2764_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_2765_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_2766_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_2767_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_2768_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_2769_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_2770_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_2771_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_2772_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_2773_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_2774_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_2775_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_2776_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_2777_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_2778_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_2779_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_2780_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_2781_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_2782_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_2783_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_2784_mult__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ K2 )
        = ( times_times_nat @ N @ K2 ) )
      = ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2785_mult__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M2 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( M2 = N )
        | ( K2 = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2786_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2787_mult__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2788_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2789_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2790_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2791_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_2792_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2793_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2794_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2795_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_2796_max_Obounded__iff,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_2797_max_Obounded__iff,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_2798_max_Obounded__iff,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_2799_max_Obounded__iff,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C @ A ) ) ) ).

% max.bounded_iff
thf(fact_2800_max__bot,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ bot_bo2099793752762293965at_nat @ X3 )
      = X3 ) ).

% max_bot
thf(fact_2801_max__bot,axiom,
    ! [X3: set_o] :
      ( ( ord_max_set_o @ bot_bot_set_o @ X3 )
      = X3 ) ).

% max_bot
thf(fact_2802_max__bot,axiom,
    ! [X3: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X3 )
      = X3 ) ).

% max_bot
thf(fact_2803_max__bot,axiom,
    ! [X3: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X3 )
      = X3 ) ).

% max_bot
thf(fact_2804_max__bot,axiom,
    ! [X3: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X3 )
      = X3 ) ).

% max_bot
thf(fact_2805_max__bot2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( ord_ma7524802468073614006at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      = X3 ) ).

% max_bot2
thf(fact_2806_max__bot2,axiom,
    ! [X3: set_o] :
      ( ( ord_max_set_o @ X3 @ bot_bot_set_o )
      = X3 ) ).

% max_bot2
thf(fact_2807_max__bot2,axiom,
    ! [X3: set_nat] :
      ( ( ord_max_set_nat @ X3 @ bot_bot_set_nat )
      = X3 ) ).

% max_bot2
thf(fact_2808_max__bot2,axiom,
    ! [X3: set_int] :
      ( ( ord_max_set_int @ X3 @ bot_bot_set_int )
      = X3 ) ).

% max_bot2
thf(fact_2809_max__bot2,axiom,
    ! [X3: nat] :
      ( ( ord_max_nat @ X3 @ bot_bot_nat )
      = X3 ) ).

% max_bot2
thf(fact_2810_max__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ M2 @ N ) ) ) ).

% max_Suc_Suc
thf(fact_2811_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_2812_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_2813_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_2814_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_2815_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_2816_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_2817_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_2818_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_2819_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_2820_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_2821_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_2822_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_2823_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_2824_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_2825_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_2826_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2827_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_2828_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_2829_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_2830_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2831_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_2832_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_2833_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_2834_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_2835_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_2836_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_2837_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_2838_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_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_2846_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2847_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_2848_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_2849_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_2850_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2851_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_2852_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_2853_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_2854_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_2855_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_2856_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_2857_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_2858_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_2859_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_2860_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_2861_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2862_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2863_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_2864_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2865_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2866_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_2867_sum__squares__eq__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2868_sum__squares__eq__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2869_sum__squares__eq__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2870_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_2871_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2872_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2873_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2874_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2875_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2876_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_2877_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_2878_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_2879_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_2880_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_2881_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2882_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2883_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2884_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2885_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_2886_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_2887_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V2 ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V2 ) )
          = ( numeral_numeral_real @ V2 ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V2 ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V2 ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_2888_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V2 ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V2 ) )
          = ( numeral_numeral_rat @ V2 ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V2 ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( numeral_numeral_rat @ V2 ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_2889_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
          = ( numeral_numeral_nat @ V2 ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
       => ( ( ord_max_nat @ ( numeral_numeral_nat @ U ) @ ( numeral_numeral_nat @ V2 ) )
          = ( numeral_numeral_nat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_2890_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
          = ( numeral_numeral_int @ V2 ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( numeral_numeral_int @ V2 ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_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_2898_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_2899_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_2900_one__eq__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M2 @ N ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2901_mult__eq__1__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( times_times_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2902_div__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = M2 ) ).

% div_by_Suc_0
thf(fact_2903_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_2904_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_2905_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_2906_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_2907_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_2908_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_2909_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_2910_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_2911_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_2912_nat__0__less__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2913_mult__less__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% mult_less_cancel2
thf(fact_2914_nat__power__eq__Suc__0__iff,axiom,
    ! [X3: nat,M2: nat] :
      ( ( ( power_power_nat @ X3 @ M2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2 = zero_zero_nat )
        | ( X3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2915_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2916_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_2917_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_2918_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_2919_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_2920_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2921_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2922_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2923_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2924_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2925_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2926_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2927_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2928_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2929_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2930_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2931_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2932_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2933_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2934_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_2935_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_2936_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_2937_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_2938_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_2939_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_2940_one__le__mult__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).

% one_le_mult_iff
thf(fact_2941_mult__le__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N @ K2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% mult_le_cancel2
thf(fact_2942_nat__mult__le__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2943_power__strict__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2944_power__strict__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2945_power__strict__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2946_power__strict__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2947_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_2948_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_2949_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_2950_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_2951_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_2952_power__mono__iff,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2953_power__mono__iff,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2954_power__mono__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2955_power__mono__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2956_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_2957_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_2958_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_2959_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_2960_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_2961_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_2962_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_2963_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_2964_power2__eq__iff__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2965_power2__eq__iff__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2966_power2__eq__iff__nonneg,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2967_power2__eq__iff__nonneg,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2968_power__decreasing__iff,axiom,
    ! [B: real,M2: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M2 ) @ ( power_power_real @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2969_power__decreasing__iff,axiom,
    ! [B: rat,M2: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M2 ) @ ( power_power_rat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2970_power__decreasing__iff,axiom,
    ! [B: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2971_power__decreasing__iff,axiom,
    ! [B: int,M2: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2972_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_2973_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_2974_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_2975_sum__power2__eq__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2976_sum__power2__eq__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2977_sum__power2__eq__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2978_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_2979_zero__reorient,axiom,
    ! [X3: literal] :
      ( ( zero_zero_literal = X3 )
      = ( X3 = zero_zero_literal ) ) ).

% zero_reorient
thf(fact_2980_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
      = ( X3 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_2981_zero__reorient,axiom,
    ! [X3: rat] :
      ( ( zero_zero_rat = X3 )
      = ( X3 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_2982_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_2983_zero__reorient,axiom,
    ! [X3: int] :
      ( ( zero_zero_int = X3 )
      = ( X3 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_2984_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_2985_max_Omono,axiom,
    ! [C: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2986_max_Omono,axiom,
    ! [C: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2987_max_Omono,axiom,
    ! [C: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2988_max_Omono,axiom,
    ! [C: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_2989_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_2990_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_2991_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_2992_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_2993_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_2994_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_2995_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_2996_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_2997_max_OboundedE,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_2998_max_OboundedE,axiom,
    ! [B: num,C: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C @ A ) ) ) ).

% max.boundedE
thf(fact_2999_max_OboundedE,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3000_max_OboundedE,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C @ A ) ) ) ).

% max.boundedE
thf(fact_3001_max_OboundedI,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3002_max_OboundedI,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3003_max_OboundedI,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3004_max_OboundedI,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C ) @ A ) ) ) ).

% max.boundedI
thf(fact_3005_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( A6
          = ( ord_max_rat @ A6 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_3006_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A6: num] :
          ( A6
          = ( ord_max_num @ A6 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_3007_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( A6
          = ( ord_max_nat @ A6 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_3008_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( A6
          = ( ord_max_int @ A6 @ B7 ) ) ) ) ).

% max.order_iff
thf(fact_3009_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3010_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_3011_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_3012_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_3013_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3014_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_3015_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_3016_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_3017_le__max__iff__disj,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z2 @ ( ord_max_rat @ X3 @ Y ) )
      = ( ( ord_less_eq_rat @ Z2 @ X3 )
        | ( ord_less_eq_rat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3018_le__max__iff__disj,axiom,
    ! [Z2: num,X3: num,Y: num] :
      ( ( ord_less_eq_num @ Z2 @ ( ord_max_num @ X3 @ Y ) )
      = ( ( ord_less_eq_num @ Z2 @ X3 )
        | ( ord_less_eq_num @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3019_le__max__iff__disj,axiom,
    ! [Z2: nat,X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X3 @ Y ) )
      = ( ( ord_less_eq_nat @ Z2 @ X3 )
        | ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3020_le__max__iff__disj,axiom,
    ! [Z2: int,X3: int,Y: int] :
      ( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X3 @ Y ) )
      = ( ( ord_less_eq_int @ Z2 @ X3 )
        | ( ord_less_eq_int @ Z2 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_3021_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( ord_max_rat @ A6 @ B7 )
          = A6 ) ) ) ).

% max.absorb_iff1
thf(fact_3022_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B7: num,A6: num] :
          ( ( ord_max_num @ A6 @ B7 )
          = A6 ) ) ) ).

% max.absorb_iff1
thf(fact_3023_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( ord_max_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% max.absorb_iff1
thf(fact_3024_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( ( ord_max_int @ A6 @ B7 )
          = A6 ) ) ) ).

% max.absorb_iff1
thf(fact_3025_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( ord_max_rat @ A6 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_3026_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A6: num,B7: num] :
          ( ( ord_max_num @ A6 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_3027_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( ord_max_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_3028_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] :
          ( ( ord_max_int @ A6 @ B7 )
          = B7 ) ) ) ).

% max.absorb_iff2
thf(fact_3029_max_OcoboundedI1,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ A )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3030_max_OcoboundedI1,axiom,
    ! [C: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C @ A )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3031_max_OcoboundedI1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ A )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3032_max_OcoboundedI1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ A )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_3033_max_OcoboundedI2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ B )
     => ( ord_less_eq_rat @ C @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3034_max_OcoboundedI2,axiom,
    ! [C: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C @ B )
     => ( ord_less_eq_num @ C @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3035_max_OcoboundedI2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C @ B )
     => ( ord_less_eq_nat @ C @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3036_max_OcoboundedI2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C @ B )
     => ( ord_less_eq_int @ C @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_3037_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def
thf(fact_3038_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A6: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def
thf(fact_3039_max__def,axiom,
    ( ord_max_num
    = ( ^ [A6: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def
thf(fact_3040_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A6: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def
thf(fact_3041_max__def,axiom,
    ( ord_max_int
    = ( ^ [A6: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def
thf(fact_3042_max__absorb1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( ord_max_set_nat @ X3 @ Y )
        = X3 ) ) ).

% max_absorb1
thf(fact_3043_max__absorb1,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ( ord_max_rat @ X3 @ Y )
        = X3 ) ) ).

% max_absorb1
thf(fact_3044_max__absorb1,axiom,
    ! [Y: num,X3: num] :
      ( ( ord_less_eq_num @ Y @ X3 )
     => ( ( ord_max_num @ X3 @ Y )
        = X3 ) ) ).

% max_absorb1
thf(fact_3045_max__absorb1,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_max_nat @ X3 @ Y )
        = X3 ) ) ).

% max_absorb1
thf(fact_3046_max__absorb1,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ( ( ord_max_int @ X3 @ Y )
        = X3 ) ) ).

% max_absorb1
thf(fact_3047_max__absorb2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_max_set_nat @ X3 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3048_max__absorb2,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_max_rat @ X3 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3049_max__absorb2,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_num @ X3 @ Y )
     => ( ( ord_max_num @ X3 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3050_max__absorb2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_max_nat @ X3 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3051_max__absorb2,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_max_int @ X3 @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_3052_max__add__distrib__left,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X3 @ Y ) @ Z2 )
      = ( ord_max_real @ ( plus_plus_real @ X3 @ Z2 ) @ ( plus_plus_real @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_3053_max__add__distrib__left,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X3 @ Y ) @ Z2 )
      = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Z2 ) @ ( plus_plus_rat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_3054_max__add__distrib__left,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X3 @ Y ) @ Z2 )
      = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Z2 ) @ ( plus_plus_nat @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_3055_max__add__distrib__left,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X3 @ Y ) @ Z2 )
      = ( ord_max_int @ ( plus_plus_int @ X3 @ Z2 ) @ ( plus_plus_int @ Y @ Z2 ) ) ) ).

% max_add_distrib_left
thf(fact_3056_max__add__distrib__right,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( plus_plus_real @ X3 @ ( ord_max_real @ Y @ Z2 ) )
      = ( ord_max_real @ ( plus_plus_real @ X3 @ Y ) @ ( plus_plus_real @ X3 @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_3057_max__add__distrib__right,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( plus_plus_rat @ X3 @ ( ord_max_rat @ Y @ Z2 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( plus_plus_rat @ X3 @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_3058_max__add__distrib__right,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( plus_plus_nat @ X3 @ ( ord_max_nat @ Y @ Z2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X3 @ Y ) @ ( plus_plus_nat @ X3 @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_3059_max__add__distrib__right,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( plus_plus_int @ X3 @ ( ord_max_int @ Y @ Z2 ) )
      = ( ord_max_int @ ( plus_plus_int @ X3 @ Y ) @ ( plus_plus_int @ X3 @ Z2 ) ) ) ).

% max_add_distrib_right
thf(fact_3060_max__diff__distrib__left,axiom,
    ! [X3: real,Y: real,Z2: real] :
      ( ( minus_minus_real @ ( ord_max_real @ X3 @ Y ) @ Z2 )
      = ( ord_max_real @ ( minus_minus_real @ X3 @ Z2 ) @ ( minus_minus_real @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_3061_max__diff__distrib__left,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X3 @ Y ) @ Z2 )
      = ( ord_max_rat @ ( minus_minus_rat @ X3 @ Z2 ) @ ( minus_minus_rat @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_3062_max__diff__distrib__left,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X3 @ Y ) @ Z2 )
      = ( ord_max_int @ ( minus_minus_int @ X3 @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).

% max_diff_distrib_left
thf(fact_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_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_3073_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_3074_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_3075_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_3076_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_3077_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_3078_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_3079_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_3080_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_3081_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_3082_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_3083_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_3084_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_3085_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_3086_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_3087_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_3088_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_3089_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_3090_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_3091_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_3092_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_3093_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_3094_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_3095_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_3096_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_3097_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_3098_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_3099_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3100_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3101_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_3102_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z: real] : ( Y5 = Z ) )
    = ( ^ [A6: real,B7: real] :
          ( ( minus_minus_real @ A6 @ B7 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3103_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z: rat] : ( Y5 = Z ) )
    = ( ^ [A6: rat,B7: rat] :
          ( ( minus_minus_rat @ A6 @ B7 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3104_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    = ( ^ [A6: int,B7: int] :
          ( ( minus_minus_int @ A6 @ B7 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3105_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_3106_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_3107_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_3108_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_3109_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_3110_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_3111_vebt__buildup_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( X3
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va3: nat] :
              ( X3
             != ( suc @ ( suc @ Va3 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_3112_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N2: nat] :
            ( X3
           != ( suc @ N2 ) ) ) ).

% list_decode.cases
thf(fact_3113_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_3114_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_3115_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_3116_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_3117_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_3118_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_3119_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: 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 @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_3120_zero__induct,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( P @ K2 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_3121_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_3122_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_3123_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_3124_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% not0_implies_Suc
thf(fact_3125_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N2 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_3126_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_3127_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_3128_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_3129_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_3130_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_3131_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_3132_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_3133_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_3134_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_3135_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_3136_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_3137_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_3138_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_3139_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_3140_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_3141_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N )
              = ( power_power_real @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3142_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N )
              = ( power_power_rat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3143_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N )
              = ( power_power_nat @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3144_power__eq__iff__eq__base,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N )
              = ( power_power_int @ B @ N ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_3145_power__eq__imp__eq__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ N )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3146_power__eq__imp__eq__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N )
        = ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3147_power__eq__imp__eq__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N )
        = ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3148_power__eq__imp__eq__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ N )
        = ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_3149_nat__minus__add__max,axiom,
    ! [N: nat,M2: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N @ M2 ) @ M2 )
      = ( ord_max_nat @ N @ M2 ) ) ).

% nat_minus_add_max
thf(fact_3150_power__strict__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3151_power__strict__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3152_power__strict__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3153_power__strict__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).

% power_strict_mono
thf(fact_3154_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_3155_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_3156_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_3157_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_3158_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_3159_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_3160_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_3161_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_3162_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3163_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3164_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3165_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_3166_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3167_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3168_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_3169_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3170_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3171_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3172_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_3173_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3174_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3175_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3176_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_3177_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3178_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3179_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3180_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_3181_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3182_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3183_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3184_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_3185_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_3186_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_3187_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_3188_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_3189_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3190_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3191_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_3192_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3193_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3194_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3195_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_3196_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3197_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3198_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_3199_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3200_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3201_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3202_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_3203_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3204_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3205_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_3206_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3207_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3208_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_3209_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3210_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3211_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_3212_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_3213_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_3214_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_3215_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3216_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3217_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3218_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_3219_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3220_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3221_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3222_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_3223_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_3224_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_3225_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_3226_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_3227_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_3228_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_3229_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_3230_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_3231_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3232_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3233_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3234_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_3235_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3236_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3237_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3238_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_3239_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3240_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3241_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3242_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_3243_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3244_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3245_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3246_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_3247_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3248_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3249_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3250_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_3251_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_3252_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3253_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_3254_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_3255_add__nonneg__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X3 @ Y )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3256_add__nonneg__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X3 @ Y )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3257_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X3 @ Y )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3258_add__nonneg__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X3 @ Y )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_3259_add__nonpos__eq__0__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X3 @ Y )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3260_add__nonpos__eq__0__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X3 @ Y )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3261_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X3 @ Y )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3262_add__nonpos__eq__0__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X3 @ Y )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_3263_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_3264_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_3265_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_3266_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_3267_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_3268_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_3269_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_3270_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_3271_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_3272_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_3273_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_3274_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_3275_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3276_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3277_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3278_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_3279_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C5: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C5 ) )
           => ( C5 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_3280_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3281_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3282_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3283_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_3284_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_3285_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_3286_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_3287_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_3288_add__less__zeroD,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_3289_add__less__zeroD,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X3 @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_3290_add__less__zeroD,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X3 @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_3291_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A6: real,B7: real] : ( ord_less_eq_real @ ( minus_minus_real @ A6 @ B7 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_3292_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A6 @ B7 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_3293_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] : ( ord_less_eq_int @ ( minus_minus_int @ A6 @ B7 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_3294_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A6: real,B7: real] : ( ord_less_real @ ( minus_minus_real @ A6 @ B7 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_3295_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A6: rat,B7: rat] : ( ord_less_rat @ ( minus_minus_rat @ A6 @ B7 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_3296_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A6: int,B7: int] : ( ord_less_int @ ( minus_minus_int @ A6 @ B7 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_3297_power__mono,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3298_power__mono,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3299_power__mono,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3300_power__mono,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono
thf(fact_3301_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_3302_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_3303_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_3304_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_3305_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_3306_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_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_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_3314_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_3315_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_3316_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_3317_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_3318_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_3319_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_3320_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M2
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_3321_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% gr0_implies_Suc
thf(fact_3322_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_3323_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_3324_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_3325_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_3326_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_3327_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_3328_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3329_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_3330_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ~ ( P @ I2 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_3331_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3332_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3333_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_3334_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I @ K )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_3335_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_3336_mult__less__mono2,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ K2 @ I ) @ ( times_times_nat @ K2 @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_3337_mult__less__mono1,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ I @ K2 ) @ ( times_times_nat @ J @ K2 ) ) ) ) ).

% mult_less_mono1
thf(fact_3338_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_3339_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_3340_nat__power__less__imp__less,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_3341_mult__eq__self__implies__10,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( times_times_nat @ M2 @ N ) )
     => ( ( N = one_one_nat )
        | ( M2 = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3342_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).

% vebt_member.simps(3)
thf(fact_3343_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_3344_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_3345_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3346_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3347_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3348_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3349_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3350_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3351_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3352_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3353_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3354_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3355_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3356_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3357_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3358_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3359_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3360_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3361_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3362_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3363_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3364_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3365_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3366_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3367_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3368_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3369_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3370_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3371_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3372_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3373_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3374_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3375_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3376_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3377_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3378_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3379_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3380_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3381_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3382_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3383_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3384_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3385_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3386_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3387_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3388_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3389_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3390_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3391_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3392_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3393_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3394_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3395_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_3396_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_3397_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_3398_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_3399_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3400_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3401_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3402_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_3403_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_3404_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_3405_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_3406_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_3407_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3408_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3409_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3410_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_3411_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3412_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3413_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3414_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_3415_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3416_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3417_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3418_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_3419_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3420_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3421_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3422_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3423_sum__squares__le__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3424_sum__squares__le__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3425_sum__squares__le__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3426_sum__squares__ge__zero,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3427_sum__squares__ge__zero,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3428_sum__squares__ge__zero,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3429_mult__left__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3430_mult__left__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3431_mult__left__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3432_mult__right__le__one__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3433_mult__right__le__one__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3434_mult__right__le__one__le,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3435_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_3436_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_3437_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3438_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3439_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_3440_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_3441_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_3442_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_3443_power__less__imp__less__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3444_power__less__imp__less__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3445_power__less__imp__less__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3446_power__less__imp__less__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_3447_sum__squares__gt__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3448_sum__squares__gt__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3449_sum__squares__gt__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3450_not__sum__squares__lt__zero,axiom,
    ! [X3: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_3451_not__sum__squares__lt__zero,axiom,
    ! [X3: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3452_not__sum__squares__lt__zero,axiom,
    ! [X3: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3453_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3454_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3455_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_3456_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_3457_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_3458_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_3459_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_3460_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_3461_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_3462_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_3463_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3464_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3465_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3466_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3467_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3468_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3469_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).

% vebt_member.simps(4)
thf(fact_3470_power__inject__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N ) )
        = ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3471_power__inject__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N ) )
        = ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3472_power__inject__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N ) )
        = ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3473_power__inject__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N ) )
        = ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_3474_power__le__imp__le__base,axiom,
    ! [A: real,N: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N ) ) @ ( power_power_real @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3475_power__le__imp__le__base,axiom,
    ! [A: rat,N: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N ) ) @ ( power_power_rat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3476_power__le__imp__le__base,axiom,
    ! [A: nat,N: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3477_power__le__imp__le__base,axiom,
    ! [A: int,N: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_3478_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_3479_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_3480_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_3481_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_3482_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X3 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst @ Smry ) ) ).

% vebt_delete.simps(5)
thf(fact_3483_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_3484_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_3485_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_3486_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_3487_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_3488_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_3489_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_3490_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_3491_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_3492_neq__if__length__neq,axiom,
    ! [Xs2: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs2 )
       != ( size_size_list_int @ Ys ) )
     => ( Xs2 != Ys ) ) ).

% neq_if_length_neq
thf(fact_3493_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I2: nat] :
                ( ( ord_less_eq_nat @ I2 @ K )
               => ~ ( P @ I2 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_3494_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_3495_one__less__mult,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% one_less_mult
thf(fact_3496_n__less__m__mult__n,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3497_n__less__n__mult__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
       => ( ord_less_nat @ N @ ( times_times_nat @ N @ M2 ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3498_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_3499_power__gt__expt,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
     => ( ord_less_nat @ K2 @ ( power_power_nat @ N @ K2 ) ) ) ).

% power_gt_expt
thf(fact_3500_nat__mult__le__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3501_div__le__mono2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K2 @ N ) @ ( divide_divide_nat @ K2 @ M2 ) ) ) ) ).

% div_le_mono2
thf(fact_3502_div__greater__zero__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ord_less_eq_nat @ N @ M2 )
        & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% div_greater_zero_iff
thf(fact_3503_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_3504_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_3505_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_3506_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_3507_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_3508_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X3 )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) ).

% vebt_delete.simps(6)
thf(fact_3509_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3510_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3511_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3512_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_3513_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_3514_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_3515_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3516_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3517_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3518_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_3519_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_3520_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_3521_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3522_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3523_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3524_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_3525_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_3526_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_3527_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3528_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3529_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3530_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_3531_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_3532_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_3533_convex__bound__le,axiom,
    ! [X3: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_eq_real @ X3 @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3534_convex__bound__le,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_eq_rat @ X3 @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3535_convex__bound__le,axiom,
    ! [X3: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_eq_int @ X3 @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3536_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3537_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3538_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3539_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3540_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_3541_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_3542_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_3543_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_3544_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_3545_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_3546_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_3547_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_3548_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_3549_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_3550_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_3551_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_3552_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3553_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3554_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3555_power__strict__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_nat @ N @ N5 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_3556_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N5 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3557_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N5 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3558_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3559_power__decreasing,axiom,
    ! [N: nat,N5: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N5 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_3560_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_3561_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_3562_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_3563_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_3564_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_3565_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_3566_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_3567_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_3568_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_3569_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_3570_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_3571_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_3572_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_3573_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3574_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3575_power__diff,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3576_power__diff,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3577_power__diff,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M2 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3578_power__diff,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3579_power__diff,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_diff
thf(fact_3580_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M5 @ N3 )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).

% div_if
thf(fact_3581_div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_geq
thf(fact_3582_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_3583_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_3584_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q3 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3585_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_3586_split__div,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ I4 ) ) ) ) ) ) ).

% split_div
thf(fact_3587_dividend__less__div__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ ( divide_divide_nat @ M2 @ N ) @ N ) ) ) ) ).

% dividend_less_div_times
thf(fact_3588_dividend__less__times__div,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ M2 @ ( plus_plus_nat @ N @ ( times_times_nat @ N @ ( divide_divide_nat @ M2 @ N ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3589_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N3 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% mult_eq_if
thf(fact_3590_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_3591_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd @ Ve ) @ Vf )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_3592_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd ) @ Ve )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_3593_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3594_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3595_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3596_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3597_convex__bound__lt,axiom,
    ! [X3: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_real @ X3 @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3598_convex__bound__lt,axiom,
    ! [X3: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_rat @ X3 @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3599_convex__bound__lt,axiom,
    ! [X3: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_int @ X3 @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3600_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_3601_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_3602_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_3603_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_3604_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_3605_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_3606_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_3607_power2__eq__imp__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3608_power2__eq__imp__eq,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3609_power2__eq__imp__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3610_power2__eq__imp__eq,axiom,
    ! [X3: int,Y: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X3 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3611_power2__le__imp__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X3 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3612_power2__le__imp__le,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X3 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3613_power2__le__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X3 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3614_power2__le__imp__le,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X3 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3615_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_3616_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_3617_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_3618_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3619_exp__add__not__zero__imp__left,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_3620_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3621_exp__add__not__zero__imp__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_3622_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: 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 @ M2 ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3623_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N: nat,M2: 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 @ M2 ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_3624_power__diff__power__eq,axiom,
    ! [A: nat,N: nat,M2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3625_power__diff__power__eq,axiom,
    ! [A: int,N: nat,M2: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_3626_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_3627_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_3628_nat__induct2,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct2
thf(fact_3629_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M5: nat] : ( if_complex @ ( M5 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3630_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M5: nat] : ( if_real @ ( M5 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3631_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M5: nat] : ( if_rat @ ( M5 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3632_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M5: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3633_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M5: nat] : ( if_int @ ( M5 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M5 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_3634_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_3635_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_3636_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_3637_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_3638_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_3639_le__div__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( divide_divide_nat @ M2 @ N )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% le_div_geq
thf(fact_3640_split__div_H,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q4: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q4 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q4 ) ) )
            & ( P @ Q4 ) ) ) ) ).

% split_div'
thf(fact_3641_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_3642_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_3643_power2__less__imp__less,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X3 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3644_power2__less__imp__less,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X3 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3645_power2__less__imp__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X3 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3646_power2__less__imp__less,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X3 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3647_sum__power2__ge__zero,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3648_sum__power2__ge__zero,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3649_sum__power2__ge__zero,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3650_sum__power2__le__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3651_sum__power2__le__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3652_sum__power2__le__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3653_not__sum__power2__lt__zero,axiom,
    ! [X3: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_3654_not__sum__power2__lt__zero,axiom,
    ! [X3: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3655_not__sum__power2__lt__zero,axiom,
    ! [X3: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_3656_sum__power2__gt__zero__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3657_sum__power2__gt__zero__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3658_sum__power2__gt__zero__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3659_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_3660_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_3661_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_3662_nat__bit__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
       => ( ! [N2: nat] :
              ( ( P @ N2 )
             => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_3663_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_3664_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_3665_subset__code_I1_J,axiom,
    ! [Xs2: list_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ B4 )
      = ( ! [X4: complex] :
            ( ( member_complex @ X4 @ ( set_complex2 @ Xs2 ) )
           => ( member_complex @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3666_subset__code_I1_J,axiom,
    ! [Xs2: list_real,B4: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs2 ) @ B4 )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Xs2 ) )
           => ( member_real @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3667_subset__code_I1_J,axiom,
    ! [Xs2: list_o,B4: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ B4 )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( member_o @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3668_subset__code_I1_J,axiom,
    ! [Xs2: list_int,B4: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ B4 )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( member_int @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3669_subset__code_I1_J,axiom,
    ! [Xs2: list_VEBT_VEBT,B4: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ B4 )
      = ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( member_VEBT_VEBT @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3670_subset__code_I1_J,axiom,
    ! [Xs2: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ B4 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( member_nat @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_3671_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_3672_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_3673_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_3674_length__induct,axiom,
    ! [P: list_int > $o,Xs2: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs2 ) ) ).

% length_induct
thf(fact_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X3: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_3682_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X3: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( 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_3683_arith__geo__mean,axiom,
    ! [U: real,X3: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X3 @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3684_arith__geo__mean,axiom,
    ! [U: rat,X3: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X3 @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3685_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_3686_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_3687_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_3688_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_3689_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K2 )
           => ? [X8: vEBT_VEBT] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ( P @ I4 @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3690_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > $o > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K2 )
           => ? [X8: $o] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ( P @ I4 @ ( nth_o @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3691_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > nat > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K2 )
           => ? [X8: nat] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ( P @ I4 @ ( nth_nat @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3692_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > int > $o] :
      ( ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ K2 )
           => ? [X8: int] : ( P @ I4 @ X8 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ( P @ I4 @ ( nth_int @ Xs @ I4 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_3693_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z: list_VEBT_VEBT] : ( Y5 = Z ) )
    = ( ^ [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_3694_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
    = ( ^ [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_3695_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z: list_nat] : ( Y5 = Z ) )
    = ( ^ [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_3696_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z: list_int] : ( Y5 = Z ) )
    = ( ^ [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_3697_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_3698_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_3699_set__update__subsetI,axiom,
    ! [Xs2: list_o,A4: set_o,X3: $o,I: nat] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A4 )
     => ( ( member_o @ X3 @ A4 )
       => ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_3700_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_3701_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_3702_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_3703_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_3704_vebt__delete_Osimps_I4_J,axiom,
    ! [Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Uu2: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) @ Uu2 )
      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList @ Summary ) ) ).

% vebt_delete.simps(4)
thf(fact_3705_vebt__member_Osimps_I2_J,axiom,
    ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) ).

% vebt_member.simps(2)
thf(fact_3706_VEBT__internal_OminNull_Osimps_I4_J,axiom,
    ! [Uw2: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux @ Uy ) ) ).

% VEBT_internal.minNull.simps(4)
thf(fact_3707_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3708_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P: $o > $o] :
      ( ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs2 ) )
           => ( P @ ( nth_o @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3709_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs2 ) )
           => ( P @ ( nth_nat @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3710_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( P @ X4 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs2 ) )
           => ( P @ ( nth_int @ Xs2 @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_3711_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_3712_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_3713_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_3714_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_3715_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_3716_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_3717_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_3718_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_3719_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_3720_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_3721_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_3722_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_3723_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_3724_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_3725_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_3726_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_3727_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_3728_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_3729_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_3730_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_3731_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_3732_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_3733_set__update__subset__insert,axiom,
    ! [Xs2: list_int,I: nat,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs2 @ I @ X3 ) ) @ ( insert_int @ X3 @ ( set_int2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3734_set__update__subset__insert,axiom,
    ! [Xs2: list_o,I: nat,X3: $o] : ( ord_less_eq_set_o @ ( set_o2 @ ( list_update_o @ Xs2 @ I @ X3 ) ) @ ( insert_o @ X3 @ ( set_o2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3735_set__update__subset__insert,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I @ X3 ) ) @ ( insert8211810215607154385at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3736_set__update__subset__insert,axiom,
    ! [Xs2: list_P5464809261938338413at_nat,I: nat,X3: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs2 @ I @ X3 ) ) @ ( insert9069300056098147895at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3737_set__update__subset__insert,axiom,
    ! [Xs2: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I @ X3 ) ) @ ( insert_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3738_set__update__subset__insert,axiom,
    ! [Xs2: list_nat,I: nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs2 @ I @ X3 ) ) @ ( insert_nat @ X3 @ ( set_nat2 @ Xs2 ) ) ) ).

% set_update_subset_insert
thf(fact_3739_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_3740_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_3741_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_3742_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_3743_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_3744_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_3745_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_3746_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_3747_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_3748_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_3749_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_3750_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_3751_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_3752_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_3753_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList: 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 ) ) @ TreeList @ 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 @ TreeList ) )
          & ~ ( ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 @ TreeList @ ( 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 ) ) @ TreeList @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_3754_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_3755_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_3756_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_3757_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_3758_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_3759_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_3760_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_3761_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_3762_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_3763_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_3764_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_3765_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_3766_vebt__pred_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] :
                  ( X3
                  = ( vEBT_Leaf @ A3 @ Uw ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va3: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va3 ) ) )
                 => ~ ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_3767_vebt__succ_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X3 @ Xa2 )
        = Y )
     => ( ! [Uu: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ Uu @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv: $o,Uw: $o] :
                ( X3
                = ( vEBT_Leaf @ Uv @ Uw ) )
           => ( ? [N2: nat] :
                  ( Xa2
                  = ( suc @ N2 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_3768_vebt__delete_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] :
                  ( X3
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [N2: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N2 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
                                & ( ~ ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                      @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va3 ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary3 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_3769_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_3770_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_3771_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_3772_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A6: $o,B7: $o] :
            ( T
            = ( vEBT_Leaf @ A6 @ B7 ) ) ) ) ).

% deg1Leaf
thf(fact_3773_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_3774_half__nonnegative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K2 ) ) ).

% half_nonnegative_int_iff
thf(fact_3775_half__negative__int__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K2 @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_3776_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_3777_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_3778_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_3779_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ X5 ) ) ) ).

% VEBT_internal.insert'.cases
thf(fact_3780_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_3781_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv2: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv2 ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_3782_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu2: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu2 @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_3783_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,Uw2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_3784_vebt__delete_Osimps_I3_J,axiom,
    ! [A: $o,B: $o,N: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ N ) ) )
      = ( vEBT_Leaf @ A @ B ) ) ).

% vebt_delete.simps(3)
thf(fact_3785_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_3786_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_3787_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A @ B ) @ X3 )
      = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X3 ) ) ).

% VEBT_internal.insert'.simps(1)
thf(fact_3788_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3789_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_3790_vebt__delete_Osimps_I2_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) )
      = ( vEBT_Leaf @ A @ $false ) ) ).

% vebt_delete.simps(2)
thf(fact_3791_VEBT__internal_OminNull_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( X3
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_3792_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X3 )
      = ( ( ( X3 = zero_zero_nat )
         => A )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => B )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_3793_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_3794_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ! [Uv: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv ) )
       => ( ! [Uu: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_3795_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X3 )
      = ( ( ( X3 = zero_zero_nat )
         => A )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => B )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_3796_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv2: $o,Uw2: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv2 @ Uw2 ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_3797_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu2: $o,Uv2: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_3798_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X3 )
     => ( ( X3
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_3799_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_3800_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_3801_real__arch__pow__inv,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ? [N2: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N2 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_3802_vebt__mint_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X3
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X3
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
       => ~ ! [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_3803_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y )
     => ( ( ( X3
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv: $o] :
                ( X3
                = ( vEBT_Leaf @ $true @ Uv ) )
           => Y )
         => ( ( ? [Uu: $o] :
                  ( X3
                  = ( vEBT_Leaf @ Uu @ $true ) )
             => Y )
           => ( ( ? [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_3804_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_3805_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 )
            & ! [Y6: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y6 )
                  & ( ( power_power_real @ Y6 @ N )
                    = A ) )
               => ( Y6 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3806_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3807_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3808_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_3809_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_3810_vebt__pred_Osimps_I2_J,axiom,
    ! [A: $o,Uw2: $o] :
      ( ( A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ Uw2 ) @ ( suc @ zero_zero_nat ) )
          = none_nat ) ) ) ).

% vebt_pred.simps(2)
thf(fact_3811_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu2: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu2 @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_3812_int__power__div__base,axiom,
    ! [M2: nat,K2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ( divide_divide_int @ ( power_power_int @ K2 @ M2 ) @ K2 )
          = ( power_power_int @ K2 @ ( minus_minus_nat @ M2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_3813_vebt__pred_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw: $o] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va3: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) )
         => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 ) )
             => ( ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
                    ( X3
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                      ( X3
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_3814_vebt__succ_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,B3: $o] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv: $o,Uw: $o,N2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) )
       => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Va2 ) )
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Ve2 ) )
           => ( ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                    ( X3
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_3815_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,S2: vEBT_VEBT,X5: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X5 ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X5 ) )
         => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_3816_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu: $o,Uv: $o,Uw: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) )
     => ( ! [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,Va2: 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 @ Va2 @ Vb2 ) @ X5 ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ X5 ) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3817_vebt__delete_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B3: $o] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,N2: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) )
         => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,Uu: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) @ Uu ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X5 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X5: nat] :
                    ( X3
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X5 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                      ( X3
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_3818_vebt__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X5: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X5 ) )
     => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X5: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X5 ) )
       => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X5: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X5 ) )
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X5: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X5 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,X5: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ X5 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_3819_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_3820_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3821_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3822_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_3823_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
             => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) ) )
                  & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
thf(fact_3824_vebt__mint_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != 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 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_3825_vebt__maxt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => ( Y != 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 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_3826_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_3827_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_3828_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_3829_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_3830_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_3831_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_3832_divide__nonneg__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_3833_divide__nonneg__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_3834_divide__nonneg__nonpos,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_3835_divide__nonneg__nonpos,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_3836_divide__nonpos__nonneg,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_3837_divide__nonpos__nonneg,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_3838_divide__nonpos__nonpos,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_3839_divide__nonpos__nonpos,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_3840_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3841_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3842_field__le__epsilon,axiom,
    ! [X3: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X3 @ Y ) ) ).

% field_le_epsilon
thf(fact_3843_field__le__epsilon,axiom,
    ! [X3: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X3 @ Y ) ) ).

% field_le_epsilon
thf(fact_3844_divide__nonpos__pos,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_3845_divide__nonpos__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_3846_divide__nonpos__neg,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3847_divide__nonpos__neg,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3848_divide__nonneg__pos,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3849_divide__nonneg__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3850_divide__nonneg__neg,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_3851_divide__nonneg__neg,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_3852_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3853_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3854_frac__less2,axiom,
    ! [X3: real,Y: real,W: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_3855_frac__less2,axiom,
    ! [X3: rat,Y: rat,W: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_3856_frac__less,axiom,
    ! [X3: real,Y: real,W: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_3857_frac__less,axiom,
    ! [X3: rat,Y: rat,W: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ X3 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_3858_frac__le,axiom,
    ! [Y: real,X3: real,W: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X3 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z2 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_3859_frac__le,axiom,
    ! [Y: rat,X3: rat,W: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X3 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z2 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_3860_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3861_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3862_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3863_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3864_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3865_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3866_add__frac__eq,axiom,
    ! [Y: complex,Z2: complex,X3: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3867_add__frac__eq,axiom,
    ! [Y: real,Z2: real,X3: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3868_add__frac__eq,axiom,
    ! [Y: rat,Z2: rat,X3: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3869_add__frac__num,axiom,
    ! [Y: complex,X3: complex,Z2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y ) @ Z2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3870_add__frac__num,axiom,
    ! [Y: real,X3: real,Z2: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y ) @ Z2 )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3871_add__frac__num,axiom,
    ! [Y: rat,X3: rat,Z2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y ) @ Z2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3872_add__num__frac,axiom,
    ! [Y: complex,Z2: complex,X3: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X3 @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3873_add__num__frac,axiom,
    ! [Y: real,Z2: real,X3: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X3 @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3874_add__num__frac,axiom,
    ! [Y: rat,Z2: rat,X3: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3875_add__divide__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3876_add__divide__eq__iff,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y @ Z2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3877_add__divide__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y @ Z2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3878_divide__add__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3879_divide__add__eq__iff,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3880_divide__add__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3881_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_3882_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_3883_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_3884_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_3885_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 ) ) ) ) )
       => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_3886_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,V: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [S2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_3887_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_3888_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,Va3: nat,TreeList2: list_VEBT_VEBT] :
              ( ? [Summary3: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_3889_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                 => ( Y
                    = ( ~ ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_3890_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Uu: $o,Uv: $o] :
            ( X3
           != ( vEBT_Leaf @ Uu @ Uv ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
               => ( ( Xa2 = Mi2 )
                  | ( Xa2 = Ma2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                 => ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
             => ~ ! [V: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Vd2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_3891_field__le__mult__one__interval,axiom,
    ! [X3: real,Y: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y ) ) )
     => ( ord_less_eq_real @ X3 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3892_field__le__mult__one__interval,axiom,
    ! [X3: rat,Y: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X3 ) @ Y ) ) )
     => ( ord_less_eq_rat @ X3 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3893_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3894_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3895_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z2: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y ) @ X3 )
       => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X3 @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3896_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z2: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y ) @ X3 )
       => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3897_mult__imp__div__pos__le,axiom,
    ! [Y: real,X3: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z2 @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3898_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X3: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z2 @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3899_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3900_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3901_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3902_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3903_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3904_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3905_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3906_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3907_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3908_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3909_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3910_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3911_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3912_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3913_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_3914_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_3915_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_3916_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_3917_frac__le__eq,axiom,
    ! [Y: real,Z2: real,X3: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y ) @ ( divide_divide_real @ W @ Z2 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3918_frac__le__eq,axiom,
    ! [Y: rat,Z2: rat,X3: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y ) @ ( divide_divide_rat @ W @ Z2 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3919_vebt__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
           => Y )
         => ( ( ? [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_3920_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 ) ) ) ) )
       => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
         => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT] :
                    ( ? [Summary3: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_3921_invar__vebt_Ocases,axiom,
    ! [A1: vEBT_VEBT,A22: nat] :
      ( ( vEBT_invar_vebt @ A1 @ A22 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A1
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A22
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) )
             => ( ( A22 = Deg2 )
               => ( ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_invar_vebt @ X @ N2 ) )
                 => ( ( vEBT_invar_vebt @ Summary3 @ M )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                     => ( ( M = N2 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N2 @ M ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                           => ~ ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A1
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( A22 = Deg2 )
                 => ( ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_invar_vebt @ X @ N2 ) )
                   => ( ( vEBT_invar_vebt @ Summary3 @ M )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                       => ( ( M
                            = ( suc @ N2 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N2 @ M ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                             => ~ ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A1
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
                 => ( ( A22 = Deg2 )
                   => ( ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_invar_vebt @ X @ N2 ) )
                     => ( ( vEBT_invar_vebt @ Summary3 @ M )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                         => ( ( M = N2 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N2 @ M ) )
                             => ( ! [I2: nat] :
                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
                                 => ( ( 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 ) ) @ M ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                    = I2 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                & ! [X: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                                                        = I2 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X )
                                                      & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A1
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary3 ) )
                   => ( ( A22 = Deg2 )
                     => ( ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ( vEBT_invar_vebt @ X @ N2 ) )
                       => ( ( vEBT_invar_vebt @ Summary3 @ M )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                           => ( ( M
                                = ( suc @ N2 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N2 @ M ) )
                               => ( ! [I2: nat] :
                                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                     => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X8 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X @ X_1 ) ) )
                                   => ( ( 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 ) ) @ M ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
                                                      = I2 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
                                                  & ! [X: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X @ N2 )
                                                          = I2 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ ( vEBT_VEBT_low @ X @ N2 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X )
                                                        & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_3922_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A12: vEBT_VEBT,A23: nat] :
          ( ( ? [A6: $o,B7: $o] :
                ( A12
                = ( vEBT_Leaf @ A6 @ B7 ) )
            & ( A23
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X8 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N3 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ N3 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N3: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A12
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N3 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N3 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
              & ( A23
                = ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ X8 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I4 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A23 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                          = I4 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N3 )
                              = I4 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N3 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_3923_vebt__insert_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => ( Y
                    = ( vEBT_Leaf @ $true @ B3 ) ) )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ $true ) ) )
                    & ( ( Xa2 != one_one_nat )
                     => ( Y
                        = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
             => ( Y
               != ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
               => ( Y
                 != ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) )
           => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                          & ~ ( ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_3924_scaling__mono,axiom,
    ! [U: real,V2: real,R2: real,S: real] :
      ( ( ord_less_eq_real @ U @ V2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
       => ( ( ord_less_eq_real @ R2 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ).

% scaling_mono
thf(fact_3925_scaling__mono,axiom,
    ! [U: rat,V2: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ).

% scaling_mono
thf(fact_3926_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_3927_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_3928_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_3929_vebt__succ_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv: $o,Uw: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ Uv @ Uw ) )
               => ! [N2: nat] :
                    ( ( Xa2
                      = ( suc @ N2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv @ Uw ) @ ( suc @ N2 ) ) ) ) ) )
           => ( ! [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux2 @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd2 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_3930_vebt__pred_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ Uw ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va3: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va3 ) ) )
                     => ( ( ( B3
                           => ( Y
                              = ( some_nat @ one_one_nat ) ) )
                          & ( ~ B3
                           => ( ( A3
                               => ( Y
                                  = ( some_nat @ zero_zero_nat ) ) )
                              & ( ~ A3
                               => ( Y = none_nat ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) )
             => ( ! [Uy2: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy2 @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Xa2 ) ) ) )
                 => ( ! [V: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X3
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( 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 @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary3 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_3931_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_3932_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_3933_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_3934_vebt__delete_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y
                    = ( vEBT_Leaf @ $false @ B3 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ( Xa2
                    = ( suc @ zero_zero_nat ) )
                 => ( ( Y
                      = ( vEBT_Leaf @ A3 @ $false ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) ) ) ) )
           => ( ! [A3: $o,B3: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N2: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N2 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N2 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT] :
                        ( ( X3
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                          ( ( X3
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
                                  & ( ~ ( ( Xa2 = Mi2 )
                                        & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                        @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va3 ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary3 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary3 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_3935_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_3936_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_3937_ex__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_3938_all__nat__less,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_3939_vebt__insert_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( ( Xa2 = zero_zero_nat )
                   => ( Y
                      = ( vEBT_Leaf @ $true @ B3 ) ) )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => ( Y
                          = ( vEBT_Leaf @ A3 @ $true ) ) )
                      & ( ( Xa2 != one_one_nat )
                       => ( Y
                          = ( 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,S2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
               => ( ( Y
                    = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ Xa2 ) ) ) )
           => ( ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ Xa2 ) ) ) )
             => ( ! [V: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                              & ~ ( ( 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 @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary3 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary3 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_3940_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X3 ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% insert'_correct
thf(fact_3941_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) ) )
                    & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
thf(fact_3942_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 ) ) ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) )
           => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_3943_vebt__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( 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 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ( ! [V: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_3944_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( ( ( 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 ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_3945_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,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_3946_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 ) ) ) ) ) )
         => ( ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Xa2 ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V ) @ TreeList2 @ S2 ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_3947_IntI,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ A4 )
     => ( ( member_complex @ C @ B4 )
       => ( member_complex @ C @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3948_IntI,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ A4 )
     => ( ( member_real @ C @ B4 )
       => ( member_real @ C @ ( inf_inf_set_real @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3949_IntI,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ A4 )
     => ( ( member_o @ C @ B4 )
       => ( member_o @ C @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3950_IntI,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ A4 )
     => ( ( member_int @ C @ B4 )
       => ( member_int @ C @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3951_IntI,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ A4 )
     => ( ( member_nat @ C @ B4 )
       => ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3952_IntI,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A4 )
     => ( ( member8440522571783428010at_nat @ C @ B4 )
       => ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% IntI
thf(fact_3953_Int__iff,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A4 @ B4 ) )
      = ( ( member_complex @ C @ A4 )
        & ( member_complex @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3954_Int__iff,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A4 @ B4 ) )
      = ( ( member_real @ C @ A4 )
        & ( member_real @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3955_Int__iff,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A4 @ B4 ) )
      = ( ( member_o @ C @ A4 )
        & ( member_o @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3956_Int__iff,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A4 @ B4 ) )
      = ( ( member_int @ C @ A4 )
        & ( member_int @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3957_Int__iff,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C @ A4 )
        & ( member_nat @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3958_Int__iff,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
      = ( ( member8440522571783428010at_nat @ C @ A4 )
        & ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_3959_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_3960_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_3961_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_3962_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_3963_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_3964_le__inf__iff,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y @ Z2 ) )
      = ( ( ord_le3146513528884898305at_nat @ X3 @ Y )
        & ( ord_le3146513528884898305at_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_3965_le__inf__iff,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_set_nat @ X3 @ Y )
        & ( ord_less_eq_set_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_3966_le__inf__iff,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ ( inf_inf_rat @ Y @ Z2 ) )
      = ( ( ord_less_eq_rat @ X3 @ Y )
        & ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_3967_le__inf__iff,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) )
      = ( ( ord_less_eq_nat @ X3 @ Y )
        & ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_3968_le__inf__iff,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ ( inf_inf_int @ Y @ Z2 ) )
      = ( ( ord_less_eq_int @ X3 @ Y )
        & ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).

% le_inf_iff
thf(fact_3969_inf__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_right
thf(fact_3970_inf__bot__right,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% inf_bot_right
thf(fact_3971_inf__bot__right,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% inf_bot_right
thf(fact_3972_inf__bot__right,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ X3 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% inf_bot_right
thf(fact_3973_inf__bot__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X3 )
      = bot_bo2099793752762293965at_nat ) ).

% inf_bot_left
thf(fact_3974_inf__bot__left,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
      = bot_bot_set_o ) ).

% inf_bot_left
thf(fact_3975_inf__bot__left,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
      = bot_bot_set_nat ) ).

% inf_bot_left
thf(fact_3976_inf__bot__left,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X3 )
      = bot_bot_set_int ) ).

% inf_bot_left
thf(fact_3977_Int__subset__iff,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
      = ( ( ord_le3146513528884898305at_nat @ C2 @ A4 )
        & ( ord_le3146513528884898305at_nat @ C2 @ B4 ) ) ) ).

% Int_subset_iff
thf(fact_3978_Int__subset__iff,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B4 ) )
      = ( ( ord_less_eq_set_nat @ C2 @ A4 )
        & ( ord_less_eq_set_nat @ C2 @ B4 ) ) ) ).

% Int_subset_iff
thf(fact_3979_Int__insert__right__if1,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
        = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3980_Int__insert__right__if1,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ A @ A4 )
     => ( ( inf_inf_set_complex @ A4 @ ( insert_complex @ A @ B4 ) )
        = ( insert_complex @ A @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3981_Int__insert__right__if1,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( member_real @ A @ A4 )
     => ( ( inf_inf_set_real @ A4 @ ( insert_real @ A @ B4 ) )
        = ( insert_real @ A @ ( inf_inf_set_real @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3982_Int__insert__right__if1,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ A @ A4 )
     => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
        = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3983_Int__insert__right__if1,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( member_int @ A @ A4 )
     => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B4 ) )
        = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3984_Int__insert__right__if1,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3985_Int__insert__right__if1,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_3986_Int__insert__right__if0,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A4 )
     => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
        = ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3987_Int__insert__right__if0,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ~ ( member_complex @ A @ A4 )
     => ( ( inf_inf_set_complex @ A4 @ ( insert_complex @ A @ B4 ) )
        = ( inf_inf_set_complex @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3988_Int__insert__right__if0,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ~ ( member_real @ A @ A4 )
     => ( ( inf_inf_set_real @ A4 @ ( insert_real @ A @ B4 ) )
        = ( inf_inf_set_real @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3989_Int__insert__right__if0,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ~ ( member_o @ A @ A4 )
     => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
        = ( inf_inf_set_o @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3990_Int__insert__right__if0,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ~ ( member_int @ A @ A4 )
     => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B4 ) )
        = ( inf_inf_set_int @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3991_Int__insert__right__if0,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
        = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3992_Int__insert__right__if0,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
        = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_3993_insert__inter__insert,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ ( insert_int @ A @ B4 ) )
      = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_3994_insert__inter__insert,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ ( insert_o @ A @ B4 ) )
      = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_3995_insert__inter__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ ( insert9069300056098147895at_nat @ A @ B4 ) )
      = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_3996_insert__inter__insert,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ ( insert_nat @ A @ B4 ) )
      = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_3997_insert__inter__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ ( insert8211810215607154385at_nat @ A @ B4 ) )
      = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_3998_Int__insert__left__if1,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ C2 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
        = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_3999_Int__insert__left__if1,axiom,
    ! [A: complex,C2: set_complex,B4: set_complex] :
      ( ( member_complex @ A @ C2 )
     => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B4 ) @ C2 )
        = ( insert_complex @ A @ ( inf_inf_set_complex @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4000_Int__insert__left__if1,axiom,
    ! [A: real,C2: set_real,B4: set_real] :
      ( ( member_real @ A @ C2 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C2 )
        = ( insert_real @ A @ ( inf_inf_set_real @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4001_Int__insert__left__if1,axiom,
    ! [A: $o,C2: set_o,B4: set_o] :
      ( ( member_o @ A @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
        = ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4002_Int__insert__left__if1,axiom,
    ! [A: int,C2: set_int,B4: set_int] :
      ( ( member_int @ A @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B4 ) @ C2 )
        = ( insert_int @ A @ ( inf_inf_set_int @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4003_Int__insert__left__if1,axiom,
    ! [A: nat,C2: set_nat,B4: set_nat] :
      ( ( member_nat @ A @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
        = ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4004_Int__insert__left__if1,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
        = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_4005_Int__insert__left__if0,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ C2 )
     => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
        = ( inf_in7913087082777306421at_nat @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4006_Int__insert__left__if0,axiom,
    ! [A: complex,C2: set_complex,B4: set_complex] :
      ( ~ ( member_complex @ A @ C2 )
     => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B4 ) @ C2 )
        = ( inf_inf_set_complex @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4007_Int__insert__left__if0,axiom,
    ! [A: real,C2: set_real,B4: set_real] :
      ( ~ ( member_real @ A @ C2 )
     => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C2 )
        = ( inf_inf_set_real @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4008_Int__insert__left__if0,axiom,
    ! [A: $o,C2: set_o,B4: set_o] :
      ( ~ ( member_o @ A @ C2 )
     => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
        = ( inf_inf_set_o @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4009_Int__insert__left__if0,axiom,
    ! [A: int,C2: set_int,B4: set_int] :
      ( ~ ( member_int @ A @ C2 )
     => ( ( inf_inf_set_int @ ( insert_int @ A @ B4 ) @ C2 )
        = ( inf_inf_set_int @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4010_Int__insert__left__if0,axiom,
    ! [A: nat,C2: set_nat,B4: set_nat] :
      ( ~ ( member_nat @ A @ C2 )
     => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
        = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4011_Int__insert__left__if0,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ C2 )
     => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) ).

% Int_insert_left_if0
thf(fact_4012_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ T3 @ ( inf_in2572325071724192079at_nat @ S3 @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_4013_Int__Un__eq_I4_J,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( sup_sup_set_nat @ T3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_4014_Int__Un__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ T3 @ ( inf_in7913087082777306421at_nat @ S3 @ T3 ) )
      = T3 ) ).

% Int_Un_eq(4)
thf(fact_4015_Int__Un__eq_I3_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ S3 @ ( inf_in2572325071724192079at_nat @ S3 @ T3 ) )
      = S3 ) ).

% Int_Un_eq(3)
thf(fact_4016_Int__Un__eq_I3_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ S3 @ ( inf_inf_set_nat @ S3 @ T3 ) )
      = S3 ) ).

% Int_Un_eq(3)
thf(fact_4017_Int__Un__eq_I3_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ S3 @ ( inf_in7913087082777306421at_nat @ S3 @ T3 ) )
      = S3 ) ).

% Int_Un_eq(3)
thf(fact_4018_Int__Un__eq_I2_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_4019_Int__Un__eq_I2_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_4020_Int__Un__eq_I2_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Int_Un_eq(2)
thf(fact_4021_Int__Un__eq_I1_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Int_Un_eq(1)
thf(fact_4022_Int__Un__eq_I1_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Int_Un_eq(1)
thf(fact_4023_Int__Un__eq_I1_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Int_Un_eq(1)
thf(fact_4024_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ T3 @ ( sup_su6327502436637775413at_nat @ S3 @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_4025_Un__Int__eq_I4_J,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( inf_inf_set_nat @ T3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_4026_Un__Int__eq_I4_J,axiom,
    ! [T3: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ T3 @ ( sup_su5525570899277871387at_nat @ S3 @ T3 ) )
      = T3 ) ).

% Un_Int_eq(4)
thf(fact_4027_Un__Int__eq_I3_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ S3 @ ( sup_su6327502436637775413at_nat @ S3 @ T3 ) )
      = S3 ) ).

% Un_Int_eq(3)
thf(fact_4028_Un__Int__eq_I3_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ S3 @ ( sup_sup_set_nat @ S3 @ T3 ) )
      = S3 ) ).

% Un_Int_eq(3)
thf(fact_4029_Un__Int__eq_I3_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ S3 @ ( sup_su5525570899277871387at_nat @ S3 @ T3 ) )
      = S3 ) ).

% Un_Int_eq(3)
thf(fact_4030_Un__Int__eq_I2_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_4031_Un__Int__eq_I2_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_4032_Un__Int__eq_I2_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S3 @ T3 ) @ T3 )
      = T3 ) ).

% Un_Int_eq(2)
thf(fact_4033_Un__Int__eq_I1_J,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Un_Int_eq(1)
thf(fact_4034_Un__Int__eq_I1_J,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Un_Int_eq(1)
thf(fact_4035_Un__Int__eq_I1_J,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,T3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ S3 @ T3 ) @ S3 )
      = S3 ) ).

% Un_Int_eq(1)
thf(fact_4036_insert__disjoint_I1_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ B4 )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( ( inf_in7913087082777306421at_nat @ A4 @ B4 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4037_insert__disjoint_I1_J,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( ( inf_inf_set_complex @ ( insert_complex @ A @ A4 ) @ B4 )
        = bot_bot_set_complex )
      = ( ~ ( member_complex @ A @ B4 )
        & ( ( inf_inf_set_complex @ A4 @ B4 )
          = bot_bot_set_complex ) ) ) ).

% insert_disjoint(1)
thf(fact_4038_insert__disjoint_I1_J,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( ( inf_inf_set_real @ ( insert_real @ A @ A4 ) @ B4 )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B4 )
        & ( ( inf_inf_set_real @ A4 @ B4 )
          = bot_bot_set_real ) ) ) ).

% insert_disjoint(1)
thf(fact_4039_insert__disjoint_I1_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4040_insert__disjoint_I1_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B4 )
        & ( ( inf_inf_set_o @ A4 @ B4 )
          = bot_bot_set_o ) ) ) ).

% insert_disjoint(1)
thf(fact_4041_insert__disjoint_I1_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B4 )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ A4 @ B4 )
          = bot_bot_set_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_4042_insert__disjoint_I1_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ B4 )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B4 )
        & ( ( inf_inf_set_int @ A4 @ B4 )
          = bot_bot_set_int ) ) ) ).

% insert_disjoint(1)
thf(fact_4043_insert__disjoint_I2_J,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) @ B4 ) )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4044_insert__disjoint_I2_J,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( bot_bot_set_complex
        = ( inf_inf_set_complex @ ( insert_complex @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_complex @ A @ B4 )
        & ( bot_bot_set_complex
          = ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4045_insert__disjoint_I2_J,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ ( insert_real @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_real @ A @ B4 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4046_insert__disjoint_I2_J,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) @ B4 ) )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4047_insert__disjoint_I2_J,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ ( insert_o @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_o @ A @ B4 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4048_insert__disjoint_I2_J,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ ( insert_nat @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_nat @ A @ B4 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4049_insert__disjoint_I2_J,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ ( insert_int @ A @ A4 ) @ B4 ) )
      = ( ~ ( member_int @ A @ B4 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_4050_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( ( inf_in7913087082777306421at_nat @ B4 @ ( insert9069300056098147895at_nat @ A @ A4 ) )
        = bot_bo228742789529271731at_nat )
      = ( ~ ( member8757157785044589968at_nat @ A @ B4 )
        & ( ( inf_in7913087082777306421at_nat @ B4 @ A4 )
          = bot_bo228742789529271731at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4051_disjoint__insert_I1_J,axiom,
    ! [B4: set_complex,A: complex,A4: set_complex] :
      ( ( ( inf_inf_set_complex @ B4 @ ( insert_complex @ A @ A4 ) )
        = bot_bot_set_complex )
      = ( ~ ( member_complex @ A @ B4 )
        & ( ( inf_inf_set_complex @ B4 @ A4 )
          = bot_bot_set_complex ) ) ) ).

% disjoint_insert(1)
thf(fact_4052_disjoint__insert_I1_J,axiom,
    ! [B4: set_real,A: real,A4: set_real] :
      ( ( ( inf_inf_set_real @ B4 @ ( insert_real @ A @ A4 ) )
        = bot_bot_set_real )
      = ( ~ ( member_real @ A @ B4 )
        & ( ( inf_inf_set_real @ B4 @ A4 )
          = bot_bot_set_real ) ) ) ).

% disjoint_insert(1)
thf(fact_4053_disjoint__insert_I1_J,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ B4 @ ( insert8211810215607154385at_nat @ A @ A4 ) )
        = bot_bo2099793752762293965at_nat )
      = ( ~ ( member8440522571783428010at_nat @ A @ B4 )
        & ( ( inf_in2572325071724192079at_nat @ B4 @ A4 )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4054_disjoint__insert_I1_J,axiom,
    ! [B4: set_o,A: $o,A4: set_o] :
      ( ( ( inf_inf_set_o @ B4 @ ( insert_o @ A @ A4 ) )
        = bot_bot_set_o )
      = ( ~ ( member_o @ A @ B4 )
        & ( ( inf_inf_set_o @ B4 @ A4 )
          = bot_bot_set_o ) ) ) ).

% disjoint_insert(1)
thf(fact_4055_disjoint__insert_I1_J,axiom,
    ! [B4: set_nat,A: nat,A4: set_nat] :
      ( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A @ A4 ) )
        = bot_bot_set_nat )
      = ( ~ ( member_nat @ A @ B4 )
        & ( ( inf_inf_set_nat @ B4 @ A4 )
          = bot_bot_set_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_4056_disjoint__insert_I1_J,axiom,
    ! [B4: set_int,A: int,A4: set_int] :
      ( ( ( inf_inf_set_int @ B4 @ ( insert_int @ A @ A4 ) )
        = bot_bot_set_int )
      = ( ~ ( member_int @ A @ B4 )
        & ( ( inf_inf_set_int @ B4 @ A4 )
          = bot_bot_set_int ) ) ) ).

% disjoint_insert(1)
thf(fact_4057_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( bot_bo228742789529271731at_nat
        = ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ B @ B4 ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A4 )
        & ( bot_bo228742789529271731at_nat
          = ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4058_disjoint__insert_I2_J,axiom,
    ! [A4: set_complex,B: complex,B4: set_complex] :
      ( ( bot_bot_set_complex
        = ( inf_inf_set_complex @ A4 @ ( insert_complex @ B @ B4 ) ) )
      = ( ~ ( member_complex @ B @ A4 )
        & ( bot_bot_set_complex
          = ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4059_disjoint__insert_I2_J,axiom,
    ! [A4: set_real,B: real,B4: set_real] :
      ( ( bot_bot_set_real
        = ( inf_inf_set_real @ A4 @ ( insert_real @ B @ B4 ) ) )
      = ( ~ ( member_real @ B @ A4 )
        & ( bot_bot_set_real
          = ( inf_inf_set_real @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4060_disjoint__insert_I2_J,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( bot_bo2099793752762293965at_nat
        = ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ B @ B4 ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A4 )
        & ( bot_bo2099793752762293965at_nat
          = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4061_disjoint__insert_I2_J,axiom,
    ! [A4: set_o,B: $o,B4: set_o] :
      ( ( bot_bot_set_o
        = ( inf_inf_set_o @ A4 @ ( insert_o @ B @ B4 ) ) )
      = ( ~ ( member_o @ B @ A4 )
        & ( bot_bot_set_o
          = ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4062_disjoint__insert_I2_J,axiom,
    ! [A4: set_nat,B: nat,B4: set_nat] :
      ( ( bot_bot_set_nat
        = ( inf_inf_set_nat @ A4 @ ( insert_nat @ B @ B4 ) ) )
      = ( ~ ( member_nat @ B @ A4 )
        & ( bot_bot_set_nat
          = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4063_disjoint__insert_I2_J,axiom,
    ! [A4: set_int,B: int,B4: set_int] :
      ( ( bot_bot_set_int
        = ( inf_inf_set_int @ A4 @ ( insert_int @ B @ B4 ) ) )
      = ( ~ ( member_int @ B @ A4 )
        & ( bot_bot_set_int
          = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_4064_Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B4 @ A4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_disjoint
thf(fact_4065_Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o @ A4 @ ( minus_minus_set_o @ B4 @ A4 ) )
      = bot_bot_set_o ) ).

% Diff_disjoint
thf(fact_4066_Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int @ A4 @ ( minus_minus_set_int @ B4 @ A4 ) )
      = bot_bot_set_int ) ).

% Diff_disjoint
thf(fact_4067_Diff__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ A4 ) )
      = bot_bot_set_nat ) ).

% Diff_disjoint
thf(fact_4068_Collect__conj__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X4: complex] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_complex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4069_Collect__conj__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_set_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4070_Collect__conj__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_list_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4071_Collect__conj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_inf_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4072_Collect__conj__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
            & ( Q @ X4 ) ) )
      = ( inf_in2572325071724192079at_nat @ ( collec3392354462482085612at_nat @ P ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_4073_Int__Collect,axiom,
    ! [X3: real,A4: set_real,P: real > $o] :
      ( ( member_real @ X3 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P ) ) )
      = ( ( member_real @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4074_Int__Collect,axiom,
    ! [X3: $o,A4: set_o,P: $o > $o] :
      ( ( member_o @ X3 @ ( inf_inf_set_o @ A4 @ ( collect_o @ P ) ) )
      = ( ( member_o @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4075_Int__Collect,axiom,
    ! [X3: int,A4: set_int,P: int > $o] :
      ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P ) ) )
      = ( ( member_int @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4076_Int__Collect,axiom,
    ! [X3: complex,A4: set_complex,P: complex > $o] :
      ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) )
      = ( ( member_complex @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4077_Int__Collect,axiom,
    ! [X3: set_nat,A4: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X3 @ ( inf_inf_set_set_nat @ A4 @ ( collect_set_nat @ P ) ) )
      = ( ( member_set_nat @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4078_Int__Collect,axiom,
    ! [X3: list_nat,A4: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X3 @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P ) ) )
      = ( ( member_list_nat @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4079_Int__Collect,axiom,
    ! [X3: nat,A4: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) )
      = ( ( member_nat @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4080_Int__Collect,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P ) ) )
      = ( ( member8440522571783428010at_nat @ X3 @ A4 )
        & ( P @ X3 ) ) ) ).

% Int_Collect
thf(fact_4081_Int__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ( ( member_real @ X4 @ A5 )
              & ( member_real @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4082_Int__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ( ( member_o @ X4 @ A5 )
              & ( member_o @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4083_Int__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ( ( member_int @ X4 @ A5 )
              & ( member_int @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4084_Int__def,axiom,
    ( inf_inf_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ( ( member_complex @ X4 @ A5 )
              & ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4085_Int__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A5 )
              & ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4086_Int__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A5 )
              & ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4087_Int__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              & ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4088_Int__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A5 )
              & ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% Int_def
thf(fact_4089_IntE,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A4 @ B4 ) )
     => ~ ( ( member_complex @ C @ A4 )
         => ~ ( member_complex @ C @ B4 ) ) ) ).

% IntE
thf(fact_4090_IntE,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A4 @ B4 ) )
     => ~ ( ( member_real @ C @ A4 )
         => ~ ( member_real @ C @ B4 ) ) ) ).

% IntE
thf(fact_4091_IntE,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A4 @ B4 ) )
     => ~ ( ( member_o @ C @ A4 )
         => ~ ( member_o @ C @ B4 ) ) ) ).

% IntE
thf(fact_4092_IntE,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A4 @ B4 ) )
     => ~ ( ( member_int @ C @ A4 )
         => ~ ( member_int @ C @ B4 ) ) ) ).

% IntE
thf(fact_4093_IntE,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
     => ~ ( ( member_nat @ C @ A4 )
         => ~ ( member_nat @ C @ B4 ) ) ) ).

% IntE
thf(fact_4094_IntE,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A4 )
         => ~ ( member8440522571783428010at_nat @ C @ B4 ) ) ) ).

% IntE
thf(fact_4095_IntD1,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A4 @ B4 ) )
     => ( member_complex @ C @ A4 ) ) ).

% IntD1
thf(fact_4096_IntD1,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A4 @ B4 ) )
     => ( member_real @ C @ A4 ) ) ).

% IntD1
thf(fact_4097_IntD1,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A4 @ B4 ) )
     => ( member_o @ C @ A4 ) ) ).

% IntD1
thf(fact_4098_IntD1,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A4 @ B4 ) )
     => ( member_int @ C @ A4 ) ) ).

% IntD1
thf(fact_4099_IntD1,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C @ A4 ) ) ).

% IntD1
thf(fact_4100_IntD1,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
     => ( member8440522571783428010at_nat @ C @ A4 ) ) ).

% IntD1
thf(fact_4101_IntD2,axiom,
    ! [C: complex,A4: set_complex,B4: set_complex] :
      ( ( member_complex @ C @ ( inf_inf_set_complex @ A4 @ B4 ) )
     => ( member_complex @ C @ B4 ) ) ).

% IntD2
thf(fact_4102_IntD2,axiom,
    ! [C: real,A4: set_real,B4: set_real] :
      ( ( member_real @ C @ ( inf_inf_set_real @ A4 @ B4 ) )
     => ( member_real @ C @ B4 ) ) ).

% IntD2
thf(fact_4103_IntD2,axiom,
    ! [C: $o,A4: set_o,B4: set_o] :
      ( ( member_o @ C @ ( inf_inf_set_o @ A4 @ B4 ) )
     => ( member_o @ C @ B4 ) ) ).

% IntD2
thf(fact_4104_IntD2,axiom,
    ! [C: int,A4: set_int,B4: set_int] :
      ( ( member_int @ C @ ( inf_inf_set_int @ A4 @ B4 ) )
     => ( member_int @ C @ B4 ) ) ).

% IntD2
thf(fact_4105_IntD2,axiom,
    ! [C: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C @ ( inf_inf_set_nat @ A4 @ B4 ) )
     => ( member_nat @ C @ B4 ) ) ).

% IntD2
thf(fact_4106_IntD2,axiom,
    ! [C: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
     => ( member8440522571783428010at_nat @ C @ B4 ) ) ).

% IntD2
thf(fact_4107_Int__assoc,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
      = ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).

% Int_assoc
thf(fact_4108_Int__assoc,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) ).

% Int_assoc
thf(fact_4109_Int__absorb,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ A4 )
      = A4 ) ).

% Int_absorb
thf(fact_4110_Int__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ A4 )
      = A4 ) ).

% Int_absorb
thf(fact_4111_Int__commute,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_4112_Int__commute,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ B5 @ A5 ) ) ) ).

% Int_commute
thf(fact_4113_Int__left__absorb,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ A4 @ B4 ) )
      = ( inf_inf_set_nat @ A4 @ B4 ) ) ).

% Int_left_absorb
thf(fact_4114_Int__left__absorb,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ).

% Int_left_absorb
thf(fact_4115_Int__left__commute,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
      = ( inf_inf_set_nat @ B4 @ ( inf_inf_set_nat @ A4 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_4116_Int__left__commute,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ B4 @ ( inf_in2572325071724192079at_nat @ A4 @ C2 ) ) ) ).

% Int_left_commute
thf(fact_4117_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_4118_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_4119_inf_OcoboundedI2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_4120_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_4121_inf_OcoboundedI2,axiom,
    ! [B: int,C: int,A: int] :
      ( ( ord_less_eq_int @ B @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_4122_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_4123_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_4124_inf_OcoboundedI1,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_4125_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_4126_inf_OcoboundedI1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_4127_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B7: set_Pr1261947904930325089at_nat,A6: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_4128_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B7: set_nat,A6: set_nat] :
          ( ( inf_inf_set_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_4129_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B7: rat,A6: rat] :
          ( ( inf_inf_rat @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_4130_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B7: nat,A6: nat] :
          ( ( inf_inf_nat @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_4131_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B7: int,A6: int] :
          ( ( inf_inf_int @ A6 @ B7 )
          = B7 ) ) ) ).

% inf.absorb_iff2
thf(fact_4132_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_4133_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( ( inf_inf_set_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_4134_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] :
          ( ( inf_inf_rat @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_4135_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( ( inf_inf_nat @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_4136_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] :
          ( ( inf_inf_int @ A6 @ B7 )
          = A6 ) ) ) ).

% inf.absorb_iff1
thf(fact_4137_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4138_inf_Ocobounded2,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4139_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4140_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4141_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4142_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4143_inf_Ocobounded1,axiom,
    ! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4144_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4145_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4146_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4147_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
          ( A6
          = ( inf_in2572325071724192079at_nat @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_4148_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] :
          ( A6
          = ( inf_inf_set_nat @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_4149_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A6: rat,B7: rat] :
          ( A6
          = ( inf_inf_rat @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_4150_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] :
          ( A6
          = ( inf_inf_nat @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_4151_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A6: int,B7: int] :
          ( A6
          = ( inf_inf_int @ A6 @ B7 ) ) ) ) ).

% inf.order_iff
thf(fact_4152_inf__greatest,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X3 @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X3 @ Z2 )
       => ( ord_le3146513528884898305at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_4153_inf__greatest,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( ord_less_eq_set_nat @ X3 @ Z2 )
       => ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_4154_inf__greatest,axiom,
    ! [X3: rat,Y: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( ord_less_eq_rat @ X3 @ Z2 )
       => ( ord_less_eq_rat @ X3 @ ( inf_inf_rat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_4155_inf__greatest,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_4156_inf__greatest,axiom,
    ! [X3: int,Y: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( ord_less_eq_int @ X3 @ Z2 )
       => ( ord_less_eq_int @ X3 @ ( inf_inf_int @ Y @ Z2 ) ) ) ) ).

% inf_greatest
thf(fact_4157_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_4158_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_4159_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C )
       => ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_4160_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_4161_inf_OboundedI,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_4162_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_4163_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_4164_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_4165_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_4166_inf_OboundedE,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_4167_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y @ X3 )
     => ( ( inf_in2572325071724192079at_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4168_inf__absorb2,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X3 )
     => ( ( inf_inf_set_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4169_inf__absorb2,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ( inf_inf_rat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4170_inf__absorb2,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( inf_inf_nat @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4171_inf__absorb2,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ Y @ X3 )
     => ( ( inf_inf_int @ X3 @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4172_inf__absorb1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X3 @ Y )
     => ( ( inf_in2572325071724192079at_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_4173_inf__absorb1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ( inf_inf_set_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_4174_inf__absorb1,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ( inf_inf_rat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_4175_inf__absorb1,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( inf_inf_nat @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_4176_inf__absorb1,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ( inf_inf_int @ X3 @ Y )
        = X3 ) ) ).

% inf_absorb1
thf(fact_4177_inf_Oabsorb2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ A )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4178_inf_Oabsorb2,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( inf_inf_set_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4179_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4180_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4181_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4182_inf_Oabsorb1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( inf_in2572325071724192079at_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4183_inf_Oabsorb1,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( inf_inf_set_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4184_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4185_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4186_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4187_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4188_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y3: set_nat] :
          ( ( inf_inf_set_nat @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4189_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y3: rat] :
          ( ( inf_inf_rat @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4190_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y3: nat] :
          ( ( inf_inf_nat @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4191_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y3: int] :
          ( ( inf_inf_int @ X4 @ Y3 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4192_inf__unique,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X5 @ Y4 )
             => ( ( ord_le3146513528884898305at_nat @ X5 @ Z3 )
               => ( ord_le3146513528884898305at_nat @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4193_inf__unique,axiom,
    ! [F: set_nat > set_nat > set_nat,X3: set_nat,Y: set_nat] :
      ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: set_nat,Y4: set_nat,Z3: set_nat] :
              ( ( ord_less_eq_set_nat @ X5 @ Y4 )
             => ( ( ord_less_eq_set_nat @ X5 @ Z3 )
               => ( ord_less_eq_set_nat @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_set_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4194_inf__unique,axiom,
    ! [F: rat > rat > rat,X3: rat,Y: rat] :
      ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: rat,Y4: rat] : ( ord_less_eq_rat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: rat,Y4: rat,Z3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y4 )
             => ( ( ord_less_eq_rat @ X5 @ Z3 )
               => ( ord_less_eq_rat @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_rat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4195_inf__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y: nat] :
      ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: nat,Y4: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y4 )
             => ( ( ord_less_eq_nat @ X5 @ Z3 )
               => ( ord_less_eq_nat @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4196_inf__unique,axiom,
    ! [F: int > int > int,X3: int,Y: int] :
      ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ X5 )
     => ( ! [X5: int,Y4: int] : ( ord_less_eq_int @ ( F @ X5 @ Y4 ) @ Y4 )
       => ( ! [X5: int,Y4: int,Z3: int] :
              ( ( ord_less_eq_int @ X5 @ Y4 )
             => ( ( ord_less_eq_int @ X5 @ Z3 )
               => ( ord_less_eq_int @ X5 @ ( F @ Y4 @ Z3 ) ) ) )
         => ( ( inf_inf_int @ X3 @ Y )
            = ( F @ X3 @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4197_inf_OorderI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ( ord_le3146513528884898305at_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_4198_inf_OorderI,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( A
        = ( inf_inf_set_nat @ A @ B ) )
     => ( ord_less_eq_set_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_4199_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( inf_inf_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% inf.orderI
thf(fact_4200_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_4201_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_4202_inf_OorderE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( A
        = ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4203_inf_OorderE,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( A
        = ( inf_inf_set_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4204_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( inf_inf_rat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4205_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4206_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4207_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ X3 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_4208_le__infI2,axiom,
    ! [B: set_nat,X3: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ X3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_4209_le__infI2,axiom,
    ! [B: rat,X3: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ X3 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_4210_le__infI2,axiom,
    ! [B: nat,X3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_4211_le__infI2,axiom,
    ! [B: int,X3: int,A: int] :
      ( ( ord_less_eq_int @ B @ X3 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_4212_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X3: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ X3 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_4213_le__infI1,axiom,
    ! [A: set_nat,X3: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X3 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_4214_le__infI1,axiom,
    ! [A: rat,X3: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X3 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_4215_le__infI1,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_4216_le__infI1,axiom,
    ! [A: int,X3: int,B: int] :
      ( ( ord_less_eq_int @ A @ X3 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_4217_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C )
     => ( ( ord_le3146513528884898305at_nat @ B @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_4218_inf__mono,axiom,
    ! [A: set_nat,C: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_4219_inf__mono,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_4220_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_4221_inf__mono,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C @ D ) ) ) ) ).

% inf_mono
thf(fact_4222_le__infI,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X3 @ A )
     => ( ( ord_le3146513528884898305at_nat @ X3 @ B )
       => ( ord_le3146513528884898305at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4223_le__infI,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ A )
     => ( ( ord_less_eq_set_nat @ X3 @ B )
       => ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4224_le__infI,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X3 @ A )
     => ( ( ord_less_eq_rat @ X3 @ B )
       => ( ord_less_eq_rat @ X3 @ ( inf_inf_rat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4225_le__infI,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ( ord_less_eq_nat @ X3 @ B )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4226_le__infI,axiom,
    ! [X3: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X3 @ A )
     => ( ( ord_less_eq_int @ X3 @ B )
       => ( ord_less_eq_int @ X3 @ ( inf_inf_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4227_le__infE,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X3 @ A )
         => ~ ( ord_le3146513528884898305at_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_4228_le__infE,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_set_nat @ X3 @ A )
         => ~ ( ord_less_eq_set_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_4229_le__infE,axiom,
    ! [X3: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X3 @ ( inf_inf_rat @ A @ B ) )
     => ~ ( ( ord_less_eq_rat @ X3 @ A )
         => ~ ( ord_less_eq_rat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_4230_le__infE,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X3 @ A )
         => ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_4231_le__infE,axiom,
    ! [X3: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X3 @ ( inf_inf_int @ A @ B ) )
     => ~ ( ( ord_less_eq_int @ X3 @ A )
         => ~ ( ord_less_eq_int @ X3 @ B ) ) ) ).

% le_infE
thf(fact_4232_inf__le2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_4233_inf__le2,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_4234_inf__le2,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_4235_inf__le2,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_4236_inf__le2,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X3 @ Y ) @ Y ) ).

% inf_le2
thf(fact_4237_inf__le1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_4238_inf__le1,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_4239_inf__le1,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_4240_inf__le1,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_4241_inf__le1,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X3 @ Y ) @ X3 ) ).

% inf_le1
thf(fact_4242_inf__sup__ord_I1_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_4243_inf__sup__ord_I1_J,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_4244_inf__sup__ord_I1_J,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_4245_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_4246_inf__sup__ord_I1_J,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X3 @ Y ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_4247_inf__sup__ord_I2_J,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4248_inf__sup__ord_I2_J,axiom,
    ! [X3: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4249_inf__sup__ord_I2_J,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4250_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4251_inf__sup__ord_I2_J,axiom,
    ! [X3: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X3 @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4252_disjoint__iff__not__equal,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ! [Y3: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y3 @ B4 )
               => ( X4 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4253_disjoint__iff__not__equal,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A4 )
           => ! [Y3: $o] :
                ( ( member_o @ Y3 @ B4 )
               => ( X4 = ~ Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4254_disjoint__iff__not__equal,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ! [Y3: nat] :
                ( ( member_nat @ Y3 @ B4 )
               => ( X4 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4255_disjoint__iff__not__equal,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ! [Y3: int] :
                ( ( member_int @ Y3 @ B4 )
               => ( X4 != Y3 ) ) ) ) ) ).

% disjoint_iff_not_equal
thf(fact_4256_Int__empty__right,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_right
thf(fact_4257_Int__empty__right,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ A4 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% Int_empty_right
thf(fact_4258_Int__empty__right,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% Int_empty_right
thf(fact_4259_Int__empty__right,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ A4 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% Int_empty_right
thf(fact_4260_Int__empty__left,axiom,
    ! [B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ B4 )
      = bot_bo2099793752762293965at_nat ) ).

% Int_empty_left
thf(fact_4261_Int__empty__left,axiom,
    ! [B4: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ B4 )
      = bot_bot_set_o ) ).

% Int_empty_left
thf(fact_4262_Int__empty__left,axiom,
    ! [B4: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ B4 )
      = bot_bot_set_nat ) ).

% Int_empty_left
thf(fact_4263_Int__empty__left,axiom,
    ! [B4: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ B4 )
      = bot_bot_set_int ) ).

% Int_empty_left
thf(fact_4264_disjoint__iff,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( ( inf_inf_set_complex @ A4 @ B4 )
        = bot_bot_set_complex )
      = ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A4 )
           => ~ ( member_complex @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4265_disjoint__iff,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ( ( inf_inf_set_real @ A4 @ B4 )
        = bot_bot_set_real )
      = ( ! [X4: real] :
            ( ( member_real @ X4 @ A4 )
           => ~ ( member_real @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4266_disjoint__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ A4 )
           => ~ ( member8440522571783428010at_nat @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4267_disjoint__iff,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ! [X4: $o] :
            ( ( member_o @ X4 @ A4 )
           => ~ ( member_o @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4268_disjoint__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
           => ~ ( member_nat @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4269_disjoint__iff,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ! [X4: int] :
            ( ( member_int @ X4 @ A4 )
           => ~ ( member_int @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_4270_Int__emptyI,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A4 )
         => ~ ( member_complex @ X5 @ B4 ) )
     => ( ( inf_inf_set_complex @ A4 @ B4 )
        = bot_bot_set_complex ) ) ).

% Int_emptyI
thf(fact_4271_Int__emptyI,axiom,
    ! [A4: set_real,B4: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A4 )
         => ~ ( member_real @ X5 @ B4 ) )
     => ( ( inf_inf_set_real @ A4 @ B4 )
        = bot_bot_set_real ) ) ).

% Int_emptyI
thf(fact_4272_Int__emptyI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A4 )
         => ~ ( member8440522571783428010at_nat @ X5 @ B4 ) )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat ) ) ).

% Int_emptyI
thf(fact_4273_Int__emptyI,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ~ ( member_o @ X5 @ B4 ) )
     => ( ( inf_inf_set_o @ A4 @ B4 )
        = bot_bot_set_o ) ) ).

% Int_emptyI
thf(fact_4274_Int__emptyI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A4 )
         => ~ ( member_nat @ X5 @ B4 ) )
     => ( ( inf_inf_set_nat @ A4 @ B4 )
        = bot_bot_set_nat ) ) ).

% Int_emptyI
thf(fact_4275_Int__emptyI,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A4 )
         => ~ ( member_int @ X5 @ B4 ) )
     => ( ( inf_inf_set_int @ A4 @ B4 )
        = bot_bot_set_int ) ) ).

% Int_emptyI
thf(fact_4276_Int__Collect__mono,axiom,
    ! [A4: set_real,B4: set_real,P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ A4 @ B4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_real @ ( inf_inf_set_real @ A4 @ ( collect_real @ P ) ) @ ( inf_inf_set_real @ B4 @ ( collect_real @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4277_Int__Collect__mono,axiom,
    ! [A4: set_o,B4: set_o,P: $o > $o,Q: $o > $o] :
      ( ( ord_less_eq_set_o @ A4 @ B4 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_o @ ( inf_inf_set_o @ A4 @ ( collect_o @ P ) ) @ ( inf_inf_set_o @ B4 @ ( collect_o @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4278_Int__Collect__mono,axiom,
    ! [A4: set_int,B4: set_int,P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ A4 @ B4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_int @ ( inf_inf_set_int @ A4 @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B4 @ ( collect_int @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4279_Int__Collect__mono,axiom,
    ! [A4: set_complex,B4: set_complex,P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ A4 @ B4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le211207098394363844omplex @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) @ ( inf_inf_set_complex @ B4 @ ( collect_complex @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4280_Int__Collect__mono,axiom,
    ! [A4: set_set_nat,B4: set_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ A4 @ B4 )
     => ( ! [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6893508408891458716et_nat @ ( inf_inf_set_set_nat @ A4 @ ( collect_set_nat @ P ) ) @ ( inf_inf_set_set_nat @ B4 @ ( collect_set_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4281_Int__Collect__mono,axiom,
    ! [A4: set_list_nat,B4: set_list_nat,P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ A4 @ B4 )
     => ( ! [X5: list_nat] :
            ( ( member_list_nat @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le6045566169113846134st_nat @ ( inf_inf_set_list_nat @ A4 @ ( collect_list_nat @ P ) ) @ ( inf_inf_set_list_nat @ B4 @ ( collect_list_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4282_Int__Collect__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ ( collec3392354462482085612at_nat @ P ) ) @ ( inf_in2572325071724192079at_nat @ B4 @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4283_Int__Collect__mono,axiom,
    ! [A4: set_nat,B4: set_nat,P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ( P @ X5 )
             => ( Q @ X5 ) ) )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_4284_Int__greatest,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ C2 @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ C2 @ B4 )
       => ( ord_le3146513528884898305at_nat @ C2 @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% Int_greatest
thf(fact_4285_Int__greatest,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A4 )
     => ( ( ord_less_eq_set_nat @ C2 @ B4 )
       => ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% Int_greatest
thf(fact_4286_Int__absorb2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_4287_Int__absorb2,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( inf_inf_set_nat @ A4 @ B4 )
        = A4 ) ) ).

% Int_absorb2
thf(fact_4288_Int__absorb1,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = B4 ) ) ).

% Int_absorb1
thf(fact_4289_Int__absorb1,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( inf_inf_set_nat @ A4 @ B4 )
        = B4 ) ) ).

% Int_absorb1
thf(fact_4290_Int__lower2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ B4 ) ).

% Int_lower2
thf(fact_4291_Int__lower2,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ B4 ) ).

% Int_lower2
thf(fact_4292_Int__lower1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ A4 ) ).

% Int_lower1
thf(fact_4293_Int__lower1,axiom,
    ! [A4: set_nat,B4: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ A4 ) ).

% Int_lower1
thf(fact_4294_Int__mono,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,D2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ D2 )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ ( inf_in2572325071724192079at_nat @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_4295_Int__mono,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat,D2: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ C2 )
     => ( ( ord_less_eq_set_nat @ B4 @ D2 )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).

% Int_mono
thf(fact_4296_Int__insert__right,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
          = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( inf_in7913087082777306421at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) )
          = ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4297_Int__insert__right,axiom,
    ! [A: complex,A4: set_complex,B4: set_complex] :
      ( ( ( member_complex @ A @ A4 )
       => ( ( inf_inf_set_complex @ A4 @ ( insert_complex @ A @ B4 ) )
          = ( insert_complex @ A @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) )
      & ( ~ ( member_complex @ A @ A4 )
       => ( ( inf_inf_set_complex @ A4 @ ( insert_complex @ A @ B4 ) )
          = ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4298_Int__insert__right,axiom,
    ! [A: real,A4: set_real,B4: set_real] :
      ( ( ( member_real @ A @ A4 )
       => ( ( inf_inf_set_real @ A4 @ ( insert_real @ A @ B4 ) )
          = ( insert_real @ A @ ( inf_inf_set_real @ A4 @ B4 ) ) ) )
      & ( ~ ( member_real @ A @ A4 )
       => ( ( inf_inf_set_real @ A4 @ ( insert_real @ A @ B4 ) )
          = ( inf_inf_set_real @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4299_Int__insert__right,axiom,
    ! [A: $o,A4: set_o,B4: set_o] :
      ( ( ( member_o @ A @ A4 )
       => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
          = ( insert_o @ A @ ( inf_inf_set_o @ A4 @ B4 ) ) ) )
      & ( ~ ( member_o @ A @ A4 )
       => ( ( inf_inf_set_o @ A4 @ ( insert_o @ A @ B4 ) )
          = ( inf_inf_set_o @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4300_Int__insert__right,axiom,
    ! [A: int,A4: set_int,B4: set_int] :
      ( ( ( member_int @ A @ A4 )
       => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B4 ) )
          = ( insert_int @ A @ ( inf_inf_set_int @ A4 @ B4 ) ) ) )
      & ( ~ ( member_int @ A @ A4 )
       => ( ( inf_inf_set_int @ A4 @ ( insert_int @ A @ B4 ) )
          = ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4301_Int__insert__right,axiom,
    ! [A: nat,A4: set_nat,B4: set_nat] :
      ( ( ( member_nat @ A @ A4 )
       => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) )
      & ( ~ ( member_nat @ A @ A4 )
       => ( ( inf_inf_set_nat @ A4 @ ( insert_nat @ A @ B4 ) )
          = ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4302_Int__insert__right,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( inf_in2572325071724192079at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) )
          = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_4303_Int__insert__left,axiom,
    ! [A: produc3843707927480180839at_nat,C2: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ A @ C2 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
          = ( insert9069300056098147895at_nat @ A @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ C2 )
       => ( ( inf_in7913087082777306421at_nat @ ( insert9069300056098147895at_nat @ A @ B4 ) @ C2 )
          = ( inf_in7913087082777306421at_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4304_Int__insert__left,axiom,
    ! [A: complex,C2: set_complex,B4: set_complex] :
      ( ( ( member_complex @ A @ C2 )
       => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B4 ) @ C2 )
          = ( insert_complex @ A @ ( inf_inf_set_complex @ B4 @ C2 ) ) ) )
      & ( ~ ( member_complex @ A @ C2 )
       => ( ( inf_inf_set_complex @ ( insert_complex @ A @ B4 ) @ C2 )
          = ( inf_inf_set_complex @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4305_Int__insert__left,axiom,
    ! [A: real,C2: set_real,B4: set_real] :
      ( ( ( member_real @ A @ C2 )
       => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C2 )
          = ( insert_real @ A @ ( inf_inf_set_real @ B4 @ C2 ) ) ) )
      & ( ~ ( member_real @ A @ C2 )
       => ( ( inf_inf_set_real @ ( insert_real @ A @ B4 ) @ C2 )
          = ( inf_inf_set_real @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4306_Int__insert__left,axiom,
    ! [A: $o,C2: set_o,B4: set_o] :
      ( ( ( member_o @ A @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
          = ( insert_o @ A @ ( inf_inf_set_o @ B4 @ C2 ) ) ) )
      & ( ~ ( member_o @ A @ C2 )
       => ( ( inf_inf_set_o @ ( insert_o @ A @ B4 ) @ C2 )
          = ( inf_inf_set_o @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4307_Int__insert__left,axiom,
    ! [A: int,C2: set_int,B4: set_int] :
      ( ( ( member_int @ A @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B4 ) @ C2 )
          = ( insert_int @ A @ ( inf_inf_set_int @ B4 @ C2 ) ) ) )
      & ( ~ ( member_int @ A @ C2 )
       => ( ( inf_inf_set_int @ ( insert_int @ A @ B4 ) @ C2 )
          = ( inf_inf_set_int @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4308_Int__insert__left,axiom,
    ! [A: nat,C2: set_nat,B4: set_nat] :
      ( ( ( member_nat @ A @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
          = ( insert_nat @ A @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
      & ( ~ ( member_nat @ A @ C2 )
       => ( ( inf_inf_set_nat @ ( insert_nat @ A @ B4 ) @ C2 )
          = ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4309_Int__insert__left,axiom,
    ! [A: product_prod_nat_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ A @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
          = ( insert8211810215607154385at_nat @ A @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ C2 )
       => ( ( inf_in2572325071724192079at_nat @ ( insert8211810215607154385at_nat @ A @ B4 ) @ C2 )
          = ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) ) ).

% Int_insert_left
thf(fact_4310_Un__Int__crazy,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) @ ( inf_in2572325071724192079at_nat @ C2 @ A4 ) )
      = ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) ) @ ( sup_su6327502436637775413at_nat @ C2 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_4311_Un__Int__crazy,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ B4 @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A4 ) )
      = ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ B4 @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_4312_Un__Int__crazy,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) ) @ ( inf_in7913087082777306421at_nat @ C2 @ A4 ) )
      = ( inf_in7913087082777306421at_nat @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) @ ( sup_su5525570899277871387at_nat @ C2 @ A4 ) ) ) ).

% Un_Int_crazy
thf(fact_4313_Int__Un__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ ( inf_in2572325071724192079at_nat @ A4 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_4314_Int__Un__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ A4 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_4315_Int__Un__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) @ ( inf_in7913087082777306421at_nat @ A4 @ C2 ) ) ) ).

% Int_Un_distrib
thf(fact_4316_Un__Int__distrib,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) @ ( sup_su6327502436637775413at_nat @ A4 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_4317_Un__Int__distrib,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ A4 @ B4 ) @ ( sup_sup_set_nat @ A4 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_4318_Un__Int__distrib,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ A4 @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) @ ( sup_su5525570899277871387at_nat @ A4 @ C2 ) ) ) ).

% Un_Int_distrib
thf(fact_4319_Int__Un__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) @ A4 )
      = ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B4 @ A4 ) @ ( inf_in2572325071724192079at_nat @ C2 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_4320_Int__Un__distrib2,axiom,
    ! [B4: set_nat,C2: set_nat,A4: set_nat] :
      ( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ C2 ) @ A4 )
      = ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ A4 ) @ ( inf_inf_set_nat @ C2 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_4321_Int__Un__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) @ A4 )
      = ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B4 @ A4 ) @ ( inf_in7913087082777306421at_nat @ C2 @ A4 ) ) ) ).

% Int_Un_distrib2
thf(fact_4322_Un__Int__distrib2,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) @ A4 )
      = ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ B4 @ A4 ) @ ( sup_su6327502436637775413at_nat @ C2 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_4323_Un__Int__distrib2,axiom,
    ! [B4: set_nat,C2: set_nat,A4: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B4 @ C2 ) @ A4 )
      = ( inf_inf_set_nat @ ( sup_sup_set_nat @ B4 @ A4 ) @ ( sup_sup_set_nat @ C2 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_4324_Un__Int__distrib2,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) @ A4 )
      = ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ B4 @ A4 ) @ ( sup_su5525570899277871387at_nat @ C2 @ A4 ) ) ) ).

% Un_Int_distrib2
thf(fact_4325_Int__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ C2 )
      = ( inf_in2572325071724192079at_nat @ A4 @ ( minus_1356011639430497352at_nat @ B4 @ C2 ) ) ) ).

% Int_Diff
thf(fact_4326_Int__Diff,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
      = ( inf_inf_set_nat @ A4 @ ( minus_minus_set_nat @ B4 @ C2 ) ) ) ).

% Int_Diff
thf(fact_4327_Diff__Int2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C2 ) @ B4 ) ) ).

% Diff_Int2
thf(fact_4328_Diff__Int2,axiom,
    ! [A4: set_nat,C2: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ B4 ) ) ).

% Diff_Int2
thf(fact_4329_Diff__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ).

% Diff_Diff_Int
thf(fact_4330_Diff__Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( inf_inf_set_nat @ A4 @ B4 ) ) ).

% Diff_Diff_Int
thf(fact_4331_Diff__Int__distrib,axiom,
    ! [C2: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ C2 @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ C2 @ A4 ) @ ( inf_in2572325071724192079at_nat @ C2 @ B4 ) ) ) ).

% Diff_Int_distrib
thf(fact_4332_Diff__Int__distrib,axiom,
    ! [C2: set_nat,A4: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ C2 @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ C2 @ A4 ) @ ( inf_inf_set_nat @ C2 @ B4 ) ) ) ).

% Diff_Int_distrib
thf(fact_4333_Diff__Int__distrib2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ C2 )
      = ( minus_1356011639430497352at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ C2 ) @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4334_Diff__Int__distrib2,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ C2 )
      = ( minus_minus_set_nat @ ( inf_inf_set_nat @ A4 @ C2 ) @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).

% Diff_Int_distrib2
thf(fact_4335_distrib__sup__le,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y @ Z2 ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X3 @ Y ) @ ( sup_su6327502436637775413at_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4336_distrib__sup__le,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ ( inf_in7913087082777306421at_nat @ Y @ Z2 ) ) @ ( inf_in7913087082777306421at_nat @ ( sup_su5525570899277871387at_nat @ X3 @ Y ) @ ( sup_su5525570899277871387at_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4337_distrib__sup__le,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ Z2 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X3 @ Y ) @ ( sup_sup_set_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4338_distrib__sup__le,axiom,
    ! [X3: rat,Y: rat,Z2: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X3 @ ( inf_inf_rat @ Y @ Z2 ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X3 @ Y ) @ ( sup_sup_rat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4339_distrib__sup__le,axiom,
    ! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X3 @ ( inf_inf_nat @ Y @ Z2 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X3 @ Y ) @ ( sup_sup_nat @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4340_distrib__sup__le,axiom,
    ! [X3: int,Y: int,Z2: int] : ( ord_less_eq_int @ ( sup_sup_int @ X3 @ ( inf_inf_int @ Y @ Z2 ) ) @ ( inf_inf_int @ ( sup_sup_int @ X3 @ Y ) @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).

% distrib_sup_le
thf(fact_4341_distrib__inf__le,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ ( inf_in2572325071724192079at_nat @ X3 @ Z2 ) ) @ ( inf_in2572325071724192079at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4342_distrib__inf__le,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] : ( ord_le1268244103169919719at_nat @ ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ Y ) @ ( inf_in7913087082777306421at_nat @ X3 @ Z2 ) ) @ ( inf_in7913087082777306421at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4343_distrib__inf__le,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ ( inf_inf_set_nat @ X3 @ Z2 ) ) @ ( inf_inf_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4344_distrib__inf__le,axiom,
    ! [X3: rat,Y: rat,Z2: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X3 @ Y ) @ ( inf_inf_rat @ X3 @ Z2 ) ) @ ( inf_inf_rat @ X3 @ ( sup_sup_rat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4345_distrib__inf__le,axiom,
    ! [X3: nat,Y: nat,Z2: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X3 @ Y ) @ ( inf_inf_nat @ X3 @ Z2 ) ) @ ( inf_inf_nat @ X3 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4346_distrib__inf__le,axiom,
    ! [X3: int,Y: int,Z2: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X3 @ Y ) @ ( inf_inf_int @ X3 @ Z2 ) ) @ ( inf_inf_int @ X3 @ ( sup_sup_int @ Y @ Z2 ) ) ) ).

% distrib_inf_le
thf(fact_4347_Diff__triv,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
     => ( ( minus_1356011639430497352at_nat @ A4 @ B4 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4348_Diff__triv,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B4 )
        = bot_bot_set_o )
     => ( ( minus_minus_set_o @ A4 @ B4 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4349_Diff__triv,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B4 )
        = bot_bot_set_int )
     => ( ( minus_minus_set_int @ A4 @ B4 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4350_Diff__triv,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
     => ( ( minus_minus_set_nat @ A4 @ B4 )
        = A4 ) ) ).

% Diff_triv
thf(fact_4351_Int__Diff__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Int_Diff_disjoint
thf(fact_4352_Int__Diff__disjoint,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ A4 @ B4 ) @ ( minus_minus_set_o @ A4 @ B4 ) )
      = bot_bot_set_o ) ).

% Int_Diff_disjoint
thf(fact_4353_Int__Diff__disjoint,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ A4 @ B4 ) @ ( minus_minus_set_int @ A4 @ B4 ) )
      = bot_bot_set_int ) ).

% Int_Diff_disjoint
thf(fact_4354_Int__Diff__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = bot_bot_set_nat ) ).

% Int_Diff_disjoint
thf(fact_4355_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ C2 )
        = ( inf_in2572325071724192079at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) ) )
      = ( ord_le3146513528884898305at_nat @ C2 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_4356_Un__Int__assoc__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) @ C2 )
        = ( inf_in7913087082777306421at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) ) )
      = ( ord_le1268244103169919719at_nat @ C2 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_4357_Un__Int__assoc__eq,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ C2 )
        = ( inf_inf_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) ) )
      = ( ord_less_eq_set_nat @ C2 @ A4 ) ) ).

% Un_Int_assoc_eq
thf(fact_4358_Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( sup_su6327502436637775413at_nat @ B4 @ C2 ) )
      = ( inf_in2572325071724192079at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ ( minus_1356011639430497352at_nat @ A4 @ C2 ) ) ) ).

% Diff_Un
thf(fact_4359_Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( sup_su5525570899277871387at_nat @ B4 @ C2 ) )
      = ( inf_in7913087082777306421at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) @ ( minus_3314409938677909166at_nat @ A4 @ C2 ) ) ) ).

% Diff_Un
thf(fact_4360_Diff__Un,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( sup_sup_set_nat @ B4 @ C2 ) )
      = ( inf_inf_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ C2 ) ) ) ).

% Diff_Un
thf(fact_4361_Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( inf_in2572325071724192079at_nat @ B4 @ C2 ) )
      = ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ ( minus_1356011639430497352at_nat @ A4 @ C2 ) ) ) ).

% Diff_Int
thf(fact_4362_Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,C2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A4 @ ( inf_in7913087082777306421at_nat @ B4 @ C2 ) )
      = ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) @ ( minus_3314409938677909166at_nat @ A4 @ C2 ) ) ) ).

% Diff_Int
thf(fact_4363_Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( inf_inf_set_nat @ B4 @ C2 ) )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ C2 ) ) ) ).

% Diff_Int
thf(fact_4364_Int__Diff__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4365_Int__Diff__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4366_Int__Diff__Un,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A4 @ B4 ) @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = A4 ) ).

% Int_Diff_Un
thf(fact_4367_Un__Diff__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4368_Un__Diff__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4369_Un__Diff__Int,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ A4 @ B4 ) @ ( inf_inf_set_nat @ A4 @ B4 ) )
      = A4 ) ).

% Un_Diff_Int
thf(fact_4370_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,Va3: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary3 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4371_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                   => ( ( Y
                        = ( ( Xa2 = Mi2 )
                          | ( Xa2 = Ma2 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4372_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ 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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
                   => ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 ) ) ) )
             => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                     => ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
               => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4373_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,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ Xa2 ) )
               => ~ ( ( Xa2 = Mi2 )
                    | ( Xa2 = Ma2 ) ) ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 = Mi2 )
                      | ( Xa2 = Ma2 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) )
           => ~ ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4374_atLeastAtMost__singleton__iff,axiom,
    ! [A: $o,B: $o,C: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ C @ bot_bot_set_o ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4375_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4376_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4377_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_4378_atLeastAtMost__singleton,axiom,
    ! [A: $o] :
      ( ( set_or8904488021354931149Most_o @ A @ A )
      = ( insert_o @ A @ bot_bot_set_o ) ) ).

% atLeastAtMost_singleton
thf(fact_4379_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_4380_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_4381_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_4382_atLeastatMost__empty,axiom,
    ! [B: $o,A: $o] :
      ( ( ord_less_o @ B @ A )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o ) ) ).

% atLeastatMost_empty
thf(fact_4383_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_4384_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_4385_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_4386_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_4387_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_4388_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4389_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4390_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4391_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4392_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4393_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_4394_atLeastatMost__empty__iff2,axiom,
    ! [A: $o,B: $o] :
      ( ( bot_bot_set_o
        = ( set_or8904488021354931149Most_o @ A @ B ) )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4395_atLeastatMost__empty__iff2,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( bot_bot_set_set_nat
        = ( set_or4548717258645045905et_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4396_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4397_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4398_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4399_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4400_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_4401_atLeastAtMost__iff,axiom,
    ! [I: $o,L: $o,U: $o] :
      ( ( member_o @ I @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = ( ( ord_less_eq_o @ L @ I )
        & ( ord_less_eq_o @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4402_atLeastAtMost__iff,axiom,
    ! [I: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I )
        & ( ord_less_eq_set_nat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_4403_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_4404_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_4405_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_4406_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_4407_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_4408_Icc__eq__Icc,axiom,
    ! [L: set_nat,H: set_nat,L3: set_nat,H3: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ L @ H )
        = ( set_or4548717258645045905et_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_set_nat @ L @ H )
          & ~ ( ord_less_eq_set_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4409_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4410_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4411_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4412_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4413_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_4414_atLeastatMost__empty__iff,axiom,
    ! [A: $o,B: $o] :
      ( ( ( set_or8904488021354931149Most_o @ A @ B )
        = bot_bot_set_o )
      = ( ~ ( ord_less_eq_o @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4415_atLeastatMost__empty__iff,axiom,
    ! [A: set_nat,B: set_nat] :
      ( ( ( set_or4548717258645045905et_nat @ A @ B )
        = bot_bot_set_set_nat )
      = ( ~ ( ord_less_eq_set_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4416_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4417_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4418_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4419_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4420_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_4421_inf__Int__eq2,axiom,
    ! [R: set_Pr4329608150637261639at_nat,S3: set_Pr4329608150637261639at_nat] :
      ( ( inf_in6124848772414083356_nat_o
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X4 @ Y3 ) @ ( inf_in7913087082777306421at_nat @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_4422_inf__Int__eq2,axiom,
    ! [R: set_Pr8218934625190621173um_num,S3: set_Pr8218934625190621173um_num] :
      ( ( inf_inf_num_num_o
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: num,Y3: num] : ( member7279096912039735102um_num @ ( product_Pair_num_num @ X4 @ Y3 ) @ ( inf_in305939755129712355um_num @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_4423_inf__Int__eq2,axiom,
    ! [R: set_Pr6200539531224447659at_num,S3: set_Pr6200539531224447659at_num] :
      ( ( inf_inf_nat_num_o
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: nat,Y3: num] : ( member9148766508732265716at_num @ ( product_Pair_nat_num @ X4 @ Y3 ) @ ( inf_in7510916698018314649at_num @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_4424_inf__Int__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( inf_inf_int_int_o
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ R )
        @ ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: int,Y3: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y3 ) @ ( inf_in2269163501485487111nt_int @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_4425_inf__Int__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( inf_inf_nat_nat_o
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
        @ ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S3 ) )
      = ( ^ [X4: nat,Y3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( inf_in2572325071724192079at_nat @ R @ S3 ) ) ) ) ).

% inf_Int_eq2
thf(fact_4426_inf__Int__eq,axiom,
    ! [R: set_complex,S3: set_complex] :
      ( ( inf_inf_complex_o
        @ ^ [X4: complex] : ( member_complex @ X4 @ R )
        @ ^ [X4: complex] : ( member_complex @ X4 @ S3 ) )
      = ( ^ [X4: complex] : ( member_complex @ X4 @ ( inf_inf_set_complex @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4427_inf__Int__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( inf_inf_real_o
        @ ^ [X4: real] : ( member_real @ X4 @ R )
        @ ^ [X4: real] : ( member_real @ X4 @ S3 ) )
      = ( ^ [X4: real] : ( member_real @ X4 @ ( inf_inf_set_real @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4428_inf__Int__eq,axiom,
    ! [R: set_o,S3: set_o] :
      ( ( inf_inf_o_o
        @ ^ [X4: $o] : ( member_o @ X4 @ R )
        @ ^ [X4: $o] : ( member_o @ X4 @ S3 ) )
      = ( ^ [X4: $o] : ( member_o @ X4 @ ( inf_inf_set_o @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4429_inf__Int__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( inf_inf_int_o
        @ ^ [X4: int] : ( member_int @ X4 @ R )
        @ ^ [X4: int] : ( member_int @ X4 @ S3 ) )
      = ( ^ [X4: int] : ( member_int @ X4 @ ( inf_inf_set_int @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4430_inf__Int__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( inf_inf_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( inf_inf_set_nat @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4431_inf__Int__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in5163264567034779214_nat_o
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ R )
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ S3 ) )
      = ( ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ ( inf_in2572325071724192079at_nat @ R @ S3 ) ) ) ) ).

% inf_Int_eq
thf(fact_4432_inf__set__def,axiom,
    ( inf_inf_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( inf_inf_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 )
            @ ^ [X4: real] : ( member_real @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4433_inf__set__def,axiom,
    ( inf_inf_set_o
    = ( ^ [A5: set_o,B5: set_o] :
          ( collect_o
          @ ( inf_inf_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A5 )
            @ ^ [X4: $o] : ( member_o @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4434_inf__set__def,axiom,
    ( inf_inf_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( inf_inf_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 )
            @ ^ [X4: int] : ( member_int @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4435_inf__set__def,axiom,
    ( inf_inf_set_complex
    = ( ^ [A5: set_complex,B5: set_complex] :
          ( collect_complex
          @ ( inf_inf_complex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A5 )
            @ ^ [X4: complex] : ( member_complex @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4436_inf__set__def,axiom,
    ( inf_inf_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( inf_inf_set_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 )
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4437_inf__set__def,axiom,
    ( inf_inf_set_list_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( inf_inf_list_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 )
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4438_inf__set__def,axiom,
    ( inf_inf_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( inf_inf_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4439_inf__set__def,axiom,
    ( inf_in2572325071724192079at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( inf_in5163264567034779214_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 )
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ) ).

% inf_set_def
thf(fact_4440_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M: nat] :
              ( ( P @ M )
             => ~ ! [X: nat] :
                    ( ( P @ X )
                   => ( ord_less_eq_nat @ X @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_4441_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc3368934014287244435at_num] :
      ~ ! [F2: nat > num > num,A3: nat,B3: nat,Acc: num] :
          ( X3
         != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A3 @ ( product_Pair_nat_num @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_4442_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X3: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X3
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_4443_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: rat,M2: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ M2 )
     => ( ( ord_less_eq_rat @ M2 @ U )
       => ( ( sup_sup_set_rat @ ( set_or633870826150836451st_rat @ L @ M2 ) @ ( set_or633870826150836451st_rat @ M2 @ U ) )
          = ( set_or633870826150836451st_rat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_4444_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: num,M2: num,U: num] :
      ( ( ord_less_eq_num @ L @ M2 )
     => ( ( ord_less_eq_num @ M2 @ U )
       => ( ( sup_sup_set_num @ ( set_or7049704709247886629st_num @ L @ M2 ) @ ( set_or7049704709247886629st_num @ M2 @ U ) )
          = ( set_or7049704709247886629st_num @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_4445_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: nat,M2: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ U )
       => ( ( sup_sup_set_nat @ ( set_or1269000886237332187st_nat @ L @ M2 ) @ ( set_or1269000886237332187st_nat @ M2 @ U ) )
          = ( set_or1269000886237332187st_nat @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_4446_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: int,M2: int,U: int] :
      ( ( ord_less_eq_int @ L @ M2 )
     => ( ( ord_less_eq_int @ M2 @ U )
       => ( ( sup_sup_set_int @ ( set_or1266510415728281911st_int @ L @ M2 ) @ ( set_or1266510415728281911st_int @ M2 @ U ) )
          = ( set_or1266510415728281911st_int @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_4447_ivl__disj__un__two__touch_I4_J,axiom,
    ! [L: real,M2: real,U: real] :
      ( ( ord_less_eq_real @ L @ M2 )
     => ( ( ord_less_eq_real @ M2 @ U )
       => ( ( sup_sup_set_real @ ( set_or1222579329274155063t_real @ L @ M2 ) @ ( set_or1222579329274155063t_real @ M2 @ U ) )
          = ( set_or1222579329274155063t_real @ L @ U ) ) ) ) ).

% ivl_disj_un_two_touch(4)
thf(fact_4448_atLeastAtMost__singleton_H,axiom,
    ! [A: $o,B: $o] :
      ( ( A = B )
     => ( ( set_or8904488021354931149Most_o @ A @ B )
        = ( insert_o @ A @ bot_bot_set_o ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4449_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4450_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4451_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_4452_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4453_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4454_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4455_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4456_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4457_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_4458_atLeast0__atMost__Suc,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_atMost_Suc
thf(fact_4459_Icc__eq__insert__lb__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( set_or1269000886237332187st_nat @ M2 @ N )
        = ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_4460_atLeastAtMostSuc__conv,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) )
        = ( insert_nat @ ( suc @ N ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_4461_atLeastAtMost__insertL,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( insert_nat @ M2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% atLeastAtMost_insertL
thf(fact_4462_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_4463_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ X3 @ bot_bot_set_o )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_right
thf(fact_4464_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ X3 @ bot_bot_set_nat )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_right
thf(fact_4465_boolean__algebra_Oconj__zero__right,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ X3 @ bot_bot_set_int )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_right
thf(fact_4466_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ bot_bo2099793752762293965at_nat @ X3 )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_4467_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ bot_bot_set_o @ X3 )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_zero_left
thf(fact_4468_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ bot_bot_set_nat @ X3 )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_zero_left
thf(fact_4469_boolean__algebra_Oconj__zero__left,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ bot_bot_set_int @ X3 )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_zero_left
thf(fact_4470_set__union,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( union_VEBT_VEBT @ Xs2 @ Ys ) )
      = ( sup_su6272177626956685416T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( set_VEBT_VEBT2 @ Ys ) ) ) ).

% set_union
thf(fact_4471_set__union,axiom,
    ! [Xs2: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs2 @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs2 ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_4472_set__union,axiom,
    ! [Xs2: list_P5464809261938338413at_nat,Ys: list_P5464809261938338413at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( union_4462254032241401953at_nat @ Xs2 @ Ys ) )
      = ( sup_su5525570899277871387at_nat @ ( set_Pr3765526544606949372at_nat @ Xs2 ) @ ( set_Pr3765526544606949372at_nat @ Ys ) ) ) ).

% set_union
thf(fact_4473_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y @ Z2 ) )
        = ( ord_less_eq_real @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4474_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y @ Z2 ) )
        = ( ord_less_eq_rat @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4475_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X3: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
        = ( ord_less_eq_int @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_4476_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y ) )
        = ( ord_less_eq_real @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4477_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ ( times_times_rat @ Z2 @ Y ) )
        = ( ord_less_eq_rat @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4478_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X3: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X3 ) @ ( times_times_int @ Z2 @ Y ) )
        = ( ord_less_eq_int @ X3 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_4479_divides__aux__eq,axiom,
    ! [Q3: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_4480_divides__aux__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_4481_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X4: nat,N3: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% low_def
thf(fact_4482_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N3: nat] : ( divide_divide_nat @ ( times_times_nat @ N3 @ ( suc @ N3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_4483_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_4484_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_4485_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_4486_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_4487_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_4488_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_4489_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_4490_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4491_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4492_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4493_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4494_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4495_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4496_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4497_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4498_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4499_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4500_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4501_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4502_mod__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_4503_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_4504_Suc__mod__mult__self4,axiom,
    ! [N: nat,K2: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N @ K2 ) @ M2 ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self4
thf(fact_4505_Suc__mod__mult__self3,axiom,
    ! [K2: nat,N: nat,M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K2 @ N ) @ M2 ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self3
thf(fact_4506_Suc__mod__mult__self2,axiom,
    ! [M2: nat,N: nat,K2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ N @ K2 ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self2
thf(fact_4507_Suc__mod__mult__self1,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M2 @ ( times_times_nat @ K2 @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_mod_mult_self1
thf(fact_4508_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_4509_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_4510_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_4511_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_4512_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_4513_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_4514_mod2__Suc__Suc,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_4515_Suc__times__numeral__mod__eq,axiom,
    ! [K2: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K2 )
       != one_one_nat )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ N ) ) @ ( numeral_numeral_nat @ K2 ) )
        = one_one_nat ) ) ).

% Suc_times_numeral_mod_eq
thf(fact_4516_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_4517_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_4518_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_4519_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_4520_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_4521_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_4522_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_4523_add__self__mod__2,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M2 @ M2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_4524_mod2__gr__0,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_4525_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4526_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4527_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_4528_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4529_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4530_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_4531_mod__add__cong,axiom,
    ! [A: nat,C: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A2 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B2 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4532_mod__add__cong,axiom,
    ! [A: int,C: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A2 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B2 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4533_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A2: code_integer,B: code_integer,B2: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A2 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B2 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A2 @ B2 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_4534_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4535_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4536_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_4537_power__mod,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4538_power__mod,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4539_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N ) @ B ) ) ).

% power_mod
thf(fact_4540_mod__Suc__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).

% mod_Suc_Suc_eq
thf(fact_4541_mod__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) @ N )
      = ( modulo_modulo_nat @ ( suc @ M2 ) @ N ) ) ).

% mod_Suc_eq
thf(fact_4542_mod__less__eq__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ M2 ) ).

% mod_less_eq_dividend
thf(fact_4543_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4544_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4545_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_4546_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4547_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4548_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4549_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4550_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4551_cong__exp__iff__simps_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_4552_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D4: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).

% mod_eqE
thf(fact_4553_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D4: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).

% mod_eqE
thf(fact_4554_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4555_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4556_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_4557_mod__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
          = N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M2 @ N ) )
         != N )
       => ( ( modulo_modulo_nat @ ( suc @ M2 ) @ N )
          = ( suc @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ).

% mod_Suc
thf(fact_4558_mod__induct,axiom,
    ! [P: nat > $o,N: nat,P2: nat,M2: nat] :
      ( ( P @ N )
     => ( ( ord_less_nat @ N @ P2 )
       => ( ( ord_less_nat @ M2 @ P2 )
         => ( ! [N2: nat] :
                ( ( ord_less_nat @ N2 @ P2 )
               => ( ( P @ N2 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N2 ) @ P2 ) ) ) )
           => ( P @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_4559_mod__Suc__le__divisor,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ ( suc @ N ) ) @ N ) ).

% mod_Suc_le_divisor
thf(fact_4560_le__mod__geq,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% le_mod_geq
thf(fact_4561_nat__mod__eq__iff,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_4562_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4563_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4564_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_4565_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4566_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4567_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_4568_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4569_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4570_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4571_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_4572_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_4573_cong__exp__iff__simps_I1_J,axiom,
    ! [N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_4574_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4575_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4576_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4577_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4578_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4579_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4580_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4581_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4582_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4583_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4584_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4585_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4586_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4587_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4588_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4589_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4590_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4591_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4592_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4593_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4594_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4595_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4596_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4597_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4598_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4599_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4600_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4601_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4602_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4603_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_4604_mod__le__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_le_divisor
thf(fact_4605_nat__mod__eq__lemma,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ? [Q2: nat] :
            ( X3
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_4606_mod__eq__nat2E,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ~ ! [S2: nat] :
              ( N
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_4607_mod__eq__nat1E,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q3 )
        = ( modulo_modulo_nat @ N @ Q3 ) )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ~ ! [S2: nat] :
              ( M2
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q3 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_4608_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N ) @ Q3 ) ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ).

% mod_mult2_eq
thf(fact_4609_split__mod,axiom,
    ! [P: nat > $o,M2: nat,N: nat] :
      ( ( P @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P @ M2 ) )
        & ( ( N != zero_zero_nat )
         => ! [I4: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I4 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_4610_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4611_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4612_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4613_Suc__times__mod__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M2 @ N ) ) @ M2 )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_4614_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4615_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4616_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4617_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_4618_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_4619_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_4620_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N: nat,M2: 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 ) ) @ M2 ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4621_div__exp__mod__exp__eq,axiom,
    ! [A: int,N: nat,M2: 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 ) ) @ M2 ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4622_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N: nat,M2: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4623_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4624_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4625_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4626_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) @ ( 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 @ M2 ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4627_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) @ ( 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 @ M2 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4628_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4629_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr4329608150637261639at_nat] :
      ( ( sup_su5525570899277871387at_nat @ X3 @ bot_bo228742789529271731at_nat )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4630_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( sup_su6327502436637775413at_nat @ X3 @ bot_bo2099793752762293965at_nat )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4631_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_o] :
      ( ( sup_sup_set_o @ X3 @ bot_bot_set_o )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4632_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_nat] :
      ( ( sup_sup_set_nat @ X3 @ bot_bot_set_nat )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4633_boolean__algebra_Odisj__zero__right,axiom,
    ! [X3: set_int] :
      ( ( sup_sup_set_int @ X3 @ bot_bot_set_int )
      = X3 ) ).

% boolean_algebra.disj_zero_right
thf(fact_4634_mod__double__modulus,axiom,
    ! [M2: code_integer,X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M2 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
       => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo364778990260209775nteger @ X3 @ M2 ) )
          | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4635_mod__double__modulus,axiom,
    ! [M2: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X3 @ M2 ) )
          | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4636_mod__double__modulus,axiom,
    ! [M2: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X3 @ M2 ) )
          | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4637_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4638_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4639_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4647_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4648_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4649_diff__shunt__var,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X3 @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_4650_diff__shunt__var,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( ( minus_minus_set_o @ X3 @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_4651_diff__shunt__var,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X3 @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_4652_diff__shunt__var,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X3 @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% diff_shunt_var
thf(fact_4653_verit__le__mono__div,axiom,
    ! [A4: nat,B4: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B4 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B4 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_4654_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_4655_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_4656_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_4657_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_4658_product__nth,axiom,
    ! [N: nat,Xs2: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_num @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs2 @ Ys ) @ N )
        = ( product_Pair_num_num @ ( nth_num @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4659_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_4660_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_4661_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_4662_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_4663_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_4664_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_o] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_o @ ( nth_o @ 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_4665_product__nth,axiom,
    ! [N: nat,Xs2: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs2 @ Ys ) @ N )
        = ( product_Pair_o_nat @ ( nth_o @ 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_4666_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_4667_product__nth,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_num] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr8326237132889035090at_num @ ( product_nat_num @ Xs2 @ Ys ) @ N )
        = ( product_Pair_nat_num @ ( nth_nat @ Xs2 @ ( divide_divide_nat @ N @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4668_obtain__set__succ,axiom,
    ! [X3: nat,Z2: nat,A4: set_nat,B4: set_nat] :
      ( ( ord_less_nat @ X3 @ Z2 )
     => ( ( vEBT_VEBT_max_in_set @ A4 @ Z2 )
       => ( ( finite_finite_nat @ B4 )
         => ( ( A4 = B4 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A4 @ X3 @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_4669_obtain__set__pred,axiom,
    ! [Z2: nat,X3: nat,A4: set_nat] :
      ( ( ord_less_nat @ Z2 @ X3 )
     => ( ( vEBT_VEBT_min_in_set @ A4 @ Z2 )
       => ( ( finite_finite_nat @ A4 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A4 @ X3 @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_4670_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_4671_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_4672_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_4673_pred__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_pred_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X: nat] :
              ( ( member_nat @ X @ Xs2 )
              & ( ord_less_nat @ X @ A ) ) ) ) ).

% pred_none_empty
thf(fact_4674_succ__none__empty,axiom,
    ! [Xs2: set_nat,A: nat] :
      ( ~ ? [X_12: nat] : ( vEBT_is_succ_in_set @ Xs2 @ A @ X_12 )
     => ( ( finite_finite_nat @ Xs2 )
       => ~ ? [X: nat] :
              ( ( member_nat @ X @ Xs2 )
              & ( ord_less_nat @ A @ X ) ) ) ) ).

% succ_none_empty
thf(fact_4675_verit__eq__simplify_I8_J,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( bit0 @ X2 )
        = ( bit0 @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_4676_List_Ofinite__set,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_4677_List_Ofinite__set,axiom,
    ! [Xs2: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_4678_List_Ofinite__set,axiom,
    ! [Xs2: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs2 ) ) ).

% List.finite_set
thf(fact_4679_List_Ofinite__set,axiom,
    ! [Xs2: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ).

% List.finite_set
thf(fact_4680_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_4681_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_4682_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_4683_length__product,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_4684_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_4685_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_4686_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_4687_length__product,axiom,
    ! [Xs2: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs2 @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_4688_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_4689_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_4690_zmod__numeral__Bit0,axiom,
    ! [V2: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_4691_eucl__rel__int,axiom,
    ! [K2: int,L: int] : ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K2 @ L ) @ ( modulo_modulo_int @ K2 @ L ) ) ) ).

% eucl_rel_int
thf(fact_4692_mod__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( modulo_modulo_int @ K2 @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_4693_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( Q3 = Q5 ) ) ) ).

% unique_quotient
thf(fact_4694_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R2: int,Q5: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q5 @ R4 ) )
       => ( R2 = R4 ) ) ) ).

% unique_remainder
thf(fact_4695_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N6: set_nat] :
        ? [M5: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N6 )
         => ( ord_less_eq_nat @ X4 @ M5 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_4696_finite__list,axiom,
    ! [A4: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_4697_finite__list,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_4698_finite__list,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_4699_finite__list,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs3 )
          = A4 ) ) ).

% finite_list
thf(fact_4700_finite__less__ub,axiom,
    ! [F: nat > nat,U: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( F @ N2 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_4701_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_4702_finite__lists__length__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
              & ( ( size_s5460976970255530739at_nat @ Xs )
                = N ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_4703_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_4704_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_4705_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_4706_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_4707_eucl__rel__int__by0,axiom,
    ! [K2: int] : ( eucl_rel_int @ K2 @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K2 ) ) ).

% eucl_rel_int_by0
thf(fact_4708_div__int__unique,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( divide_divide_int @ K2 @ L )
        = Q3 ) ) ).

% div_int_unique
thf(fact_4709_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_4710_finite__lists__length__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A4 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs ) @ N ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_4711_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_4712_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_4713_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_4714_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_4715_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K2
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_4716_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4717_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4718_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4719_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4720_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_4721_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4722_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4723_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4724_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_4725_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_4726_verit__comp__simplify1_I3_J,axiom,
    ! [B2: real,A2: real] :
      ( ( ~ ( ord_less_eq_real @ B2 @ A2 ) )
      = ( ord_less_real @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4727_verit__comp__simplify1_I3_J,axiom,
    ! [B2: rat,A2: rat] :
      ( ( ~ ( ord_less_eq_rat @ B2 @ A2 ) )
      = ( ord_less_rat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4728_verit__comp__simplify1_I3_J,axiom,
    ! [B2: num,A2: num] :
      ( ( ~ ( ord_less_eq_num @ B2 @ A2 ) )
      = ( ord_less_num @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4729_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4730_verit__comp__simplify1_I3_J,axiom,
    ! [B2: int,A2: int] :
      ( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
      = ( ord_less_int @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_4731_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_4732_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_4733_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_4734_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_4735_verit__eq__simplify_I10_J,axiom,
    ! [X2: num] :
      ( one
     != ( bit0 @ X2 ) ) ).

% verit_eq_simplify(10)
thf(fact_4736_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( ( K2
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4737_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_4738_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_4739_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A6: set_nat,B7: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def_raw
thf(fact_4740_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A6: rat,B7: rat] : ( if_rat @ ( ord_less_eq_rat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def_raw
thf(fact_4741_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A6: num,B7: num] : ( if_num @ ( ord_less_eq_num @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def_raw
thf(fact_4742_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A6: nat,B7: nat] : ( if_nat @ ( ord_less_eq_nat @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def_raw
thf(fact_4743_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A6: int,B7: int] : ( if_int @ ( ord_less_eq_int @ A6 @ B7 ) @ B7 @ A6 ) ) ) ).

% max_def_raw
thf(fact_4744_finite__Diff__insert,axiom,
    ! [A4: set_int,A: int,B4: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ B4 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4745_finite__Diff__insert,axiom,
    ! [A4: set_o,A: $o,B4: set_o] :
      ( ( finite_finite_o @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ B4 ) ) )
      = ( finite_finite_o @ ( minus_minus_set_o @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4746_finite__Diff__insert,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ B4 ) ) )
      = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4747_finite__Diff__insert,axiom,
    ! [A4: set_complex,A: complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ B4 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4748_finite__Diff__insert,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ B4 ) ) )
      = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4749_finite__Diff__insert,axiom,
    ! [A4: set_nat,A: nat,B4: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ B4 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ).

% finite_Diff_insert
thf(fact_4750_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N3: nat] : ( ord_less_eq_nat @ N3 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_4751_finite__Collect__subsets,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite6551019134538273531omplex
        @ ( collect_set_complex
          @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_4752_finite__Collect__subsets,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B5 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_4753_finite__Collect__subsets,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A4 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_4754_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Z4: real] :
              ( ( power_power_real @ Z4 @ N )
              = one_one_real ) ) ) ) ).

% finite_roots_unity
thf(fact_4755_finite__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = one_one_complex ) ) ) ) ).

% finite_roots_unity
thf(fact_4756_finite__induct__select,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ S3 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [T4: set_Pr4329608150637261639at_nat] :
              ( ( ord_le2604355607129572851at_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X @ ( minus_3314409938677909166at_nat @ S3 @ T4 ) )
                    & ( P @ ( insert9069300056098147895at_nat @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4757_finite__induct__select,axiom,
    ! [S3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: complex] :
                    ( ( member_complex @ X @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
                    & ( P @ ( insert_complex @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4758_finite__induct__select,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S3 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X @ ( minus_1356011639430497352at_nat @ S3 @ T4 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4759_finite__induct__select,axiom,
    ! [S3: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [T4: set_o] :
              ( ( ord_less_set_o @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: $o] :
                    ( ( member_o @ X @ ( minus_minus_set_o @ S3 @ T4 ) )
                    & ( P @ ( insert_o @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4760_finite__induct__select,axiom,
    ! [S3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: int] :
                    ( ( member_int @ X @ ( minus_minus_set_int @ S3 @ T4 ) )
                    & ( P @ ( insert_int @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4761_finite__induct__select,axiom,
    ! [S3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X: nat] :
                    ( ( member_nat @ X @ ( minus_minus_set_nat @ S3 @ T4 ) )
                    & ( P @ ( insert_nat @ X @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_4762_remove__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,B4: set_Pr4329608150637261639at_nat] :
      ( ( P @ bot_bo228742789529271731at_nat )
     => ( ( ~ ( finite4343798906461161616at_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A8 )
             => ( ( A8 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A8 @ B4 )
                 => ( ! [X: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X @ A8 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4763_remove__induct,axiom,
    ! [P: set_real > $o,B4: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B4 )
                 => ( ! [X: real] :
                        ( ( member_real @ X @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4764_remove__induct,axiom,
    ! [P: set_complex > $o,B4: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B4 )
                 => ( ! [X: complex] :
                        ( ( member_complex @ X @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4765_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B4: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A8 )
             => ( ( A8 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A8 @ B4 )
                 => ( ! [X: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X @ A8 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4766_remove__induct,axiom,
    ! [P: set_o > $o,B4: set_o] :
      ( ( P @ bot_bot_set_o )
     => ( ( ~ ( finite_finite_o @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B4 )
                 => ( ! [X: $o] :
                        ( ( member_o @ X @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4767_remove__induct,axiom,
    ! [P: set_int > $o,B4: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B4 )
                 => ( ! [X: int] :
                        ( ( member_int @ X @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4768_remove__induct,axiom,
    ! [P: set_nat > $o,B4: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B4 )
         => ( P @ B4 ) )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B4 )
                 => ( ! [X: nat] :
                        ( ( member_nat @ X @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% remove_induct
thf(fact_4769_finite__remove__induct,axiom,
    ! [B4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B4 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [A8: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A8 )
             => ( ( A8 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A8 @ B4 )
                 => ( ! [X: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X @ A8 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4770_finite__remove__induct,axiom,
    ! [B4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( A8 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A8 @ B4 )
                 => ( ! [X: real] :
                        ( ( member_real @ X @ A8 )
                       => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4771_finite__remove__induct,axiom,
    ! [B4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( A8 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A8 @ B4 )
                 => ( ! [X: complex] :
                        ( ( member_complex @ X @ A8 )
                       => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4772_finite__remove__induct,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A8: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A8 )
             => ( ( A8 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A8 @ B4 )
                 => ( ! [X: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X @ A8 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4773_finite__remove__induct,axiom,
    ! [B4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ B4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( A8 != bot_bot_set_o )
               => ( ( ord_less_eq_set_o @ A8 @ B4 )
                 => ( ! [X: $o] :
                        ( ( member_o @ X @ A8 )
                       => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ X @ bot_bot_set_o ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4774_finite__remove__induct,axiom,
    ! [B4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( A8 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A8 @ B4 )
                 => ( ! [X: int] :
                        ( ( member_int @ X @ A8 )
                       => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4775_finite__remove__induct,axiom,
    ! [B4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( A8 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A8 @ B4 )
                 => ( ! [X: nat] :
                        ( ( member_nat @ X @ A8 )
                       => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
                   => ( P @ A8 ) ) ) ) )
         => ( P @ B4 ) ) ) ) ).

% finite_remove_induct
thf(fact_4776_finite__insert,axiom,
    ! [A: int,A4: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A @ A4 ) )
      = ( finite_finite_int @ A4 ) ) ).

% finite_insert
thf(fact_4777_finite__insert,axiom,
    ! [A: $o,A4: set_o] :
      ( ( finite_finite_o @ ( insert_o @ A @ A4 ) )
      = ( finite_finite_o @ A4 ) ) ).

% finite_insert
thf(fact_4778_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) )
      = ( finite4343798906461161616at_nat @ A4 ) ) ).

% finite_insert
thf(fact_4779_finite__insert,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A4 ) )
      = ( finite_finite_nat @ A4 ) ) ).

% finite_insert
thf(fact_4780_finite__insert,axiom,
    ! [A: complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) )
      = ( finite3207457112153483333omplex @ A4 ) ) ).

% finite_insert
thf(fact_4781_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) )
      = ( finite6177210948735845034at_nat @ A4 ) ) ).

% finite_insert
thf(fact_4782_finite__maxlen,axiom,
    ! [M7: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M7 )
     => ? [N2: nat] :
        ! [X: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X @ M7 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4783_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N2: nat] :
        ! [X: list_o] :
          ( ( member_list_o @ X @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4784_finite__maxlen,axiom,
    ! [M7: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M7 )
     => ? [N2: nat] :
        ! [X: list_nat] :
          ( ( member_list_nat @ X @ M7 )
         => ( ord_less_nat @ ( size_size_list_nat @ X ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4785_finite__maxlen,axiom,
    ! [M7: set_list_int] :
      ( ( finite3922522038869484883st_int @ M7 )
     => ? [N2: nat] :
        ! [X: list_int] :
          ( ( member_list_int @ X @ M7 )
         => ( ord_less_nat @ ( size_size_list_int @ X ) @ N2 ) ) ) ).

% finite_maxlen
thf(fact_4786_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_4787_finite__has__maximal2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ A @ A4 )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A4 )
            & ( ord_less_eq_o @ A @ X5 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_4788_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X5 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_4789_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_4790_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_4791_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_4792_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_4793_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_4794_finite__has__minimal2,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ A @ A4 )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A4 )
            & ( ord_less_eq_o @ X5 @ A )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_4795_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ( ord_less_eq_set_nat @ X5 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_4796_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_4797_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_4798_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_4799_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_4800_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_4801_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_4802_finite_OemptyI,axiom,
    finite_finite_o @ bot_bot_set_o ).

% finite.emptyI
thf(fact_4803_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_4804_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_4805_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_4806_infinite__imp__nonempty,axiom,
    ! [S3: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S3 )
     => ( S3 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_4807_infinite__imp__nonempty,axiom,
    ! [S3: set_o] :
      ( ~ ( finite_finite_o @ S3 )
     => ( S3 != bot_bot_set_o ) ) ).

% infinite_imp_nonempty
thf(fact_4808_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_4809_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_4810_rev__finite__subset,axiom,
    ! [B4: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_4811_rev__finite__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_4812_rev__finite__subset,axiom,
    ! [B4: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% rev_finite_subset
thf(fact_4813_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_4814_infinite__super,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S3 @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S3 )
       => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_4815_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_4816_finite__subset,axiom,
    ! [A4: set_complex,B4: set_complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ B4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( finite3207457112153483333omplex @ A4 ) ) ) ).

% finite_subset
thf(fact_4817_finite__subset,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_4818_finite__subset,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( finite_finite_nat @ B4 )
       => ( finite_finite_nat @ A4 ) ) ) ).

% finite_subset
thf(fact_4819_finite_OinsertI,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( finite_finite_int @ ( insert_int @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4820_finite_OinsertI,axiom,
    ! [A4: set_o,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_o @ ( insert_o @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4821_finite_OinsertI,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4822_finite_OinsertI,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4823_finite_OinsertI,axiom,
    ! [A4: set_complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4824_finite_OinsertI,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A4 ) ) ) ).

% finite.insertI
thf(fact_4825_finite__has__minimal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A4 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_4826_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X5 )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_4827_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_4828_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_4829_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_4830_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_4831_finite__has__maximal,axiom,
    ! [A4: set_o] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ A4 )
            & ! [Xa: $o] :
                ( ( member_o @ Xa @ A4 )
               => ( ( ord_less_eq_o @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_4832_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X5: set_nat] :
            ( ( member_set_nat @ X5 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X5 @ Xa )
                 => ( X5 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_4833_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_4834_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_4835_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_4836_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_4837_infinite__finite__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,A4: set_Pr4329608150637261639at_nat] :
      ( ! [A8: set_Pr4329608150637261639at_nat] :
          ( ~ ( finite4343798906461161616at_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F3 )
             => ( ~ ( member8757157785044589968at_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert9069300056098147895at_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4838_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A4: set_real] :
      ( ! [A8: set_real] :
          ( ~ ( finite_finite_real @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_real @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4839_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A4: set_complex] :
      ( ! [A8: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_complex @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4840_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A4: set_Pr1261947904930325089at_nat] :
      ( ! [A8: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F3 )
             => ( ~ ( member8440522571783428010at_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4841_infinite__finite__induct,axiom,
    ! [P: set_o > $o,A4: set_o] :
      ( ! [A8: set_o] :
          ( ~ ( finite_finite_o @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_o @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4842_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A4: set_nat] :
      ( ! [A8: set_nat] :
          ( ~ ( finite_finite_nat @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4843_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A4: set_int] :
      ( ! [A8: set_int] :
          ( ~ ( finite_finite_int @ A8 )
         => ( P @ A8 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_int @ X5 @ F3 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4844_finite__ne__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( F4 != bot_bo228742789529271731at_nat )
       => ( ! [X5: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X5 @ bot_bo228742789529271731at_nat ) )
         => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F3 )
               => ( ( F3 != bot_bo228742789529271731at_nat )
                 => ( ~ ( member8757157785044589968at_nat @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert9069300056098147895at_nat @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4845_finite__ne__induct,axiom,
    ! [F4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( F4 != bot_bot_set_real )
       => ( ! [X5: real] : ( P @ ( insert_real @ X5 @ bot_bot_set_real ) )
         => ( ! [X5: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( F3 != bot_bot_set_real )
                 => ( ~ ( member_real @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_real @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4846_finite__ne__induct,axiom,
    ! [F4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( F4 != bot_bot_set_complex )
       => ( ! [X5: complex] : ( P @ ( insert_complex @ X5 @ bot_bot_set_complex ) )
         => ( ! [X5: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( F3 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_complex @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4847_finite__ne__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( F4 != bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X5 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( F3 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert8211810215607154385at_nat @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4848_finite__ne__induct,axiom,
    ! [F4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( F4 != bot_bot_set_o )
       => ( ! [X5: $o] : ( P @ ( insert_o @ X5 @ bot_bot_set_o ) )
         => ( ! [X5: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( F3 != bot_bot_set_o )
                 => ( ~ ( member_o @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_o @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4849_finite__ne__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( F4 != bot_bot_set_nat )
       => ( ! [X5: nat] : ( P @ ( insert_nat @ X5 @ bot_bot_set_nat ) )
         => ( ! [X5: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( F3 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4850_finite__ne__induct,axiom,
    ! [F4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( F4 != bot_bot_set_int )
       => ( ! [X5: int] : ( P @ ( insert_int @ X5 @ bot_bot_set_int ) )
         => ( ! [X5: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( F3 != bot_bot_set_int )
                 => ( ~ ( member_int @ X5 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_int @ X5 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4851_finite__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X5: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F3 )
             => ( ~ ( member8757157785044589968at_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert9069300056098147895at_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4852_finite__induct,axiom,
    ! [F4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,F3: set_real] :
              ( ( finite_finite_real @ F3 )
             => ( ~ ( member_real @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_real @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4853_finite__induct,axiom,
    ! [F4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,F3: set_complex] :
              ( ( finite3207457112153483333omplex @ F3 )
             => ( ~ ( member_complex @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_complex @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4854_finite__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X5: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F3 )
             => ( ~ ( member8440522571783428010at_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert8211810215607154385at_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4855_finite__induct,axiom,
    ! [F4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,F3: set_o] :
              ( ( finite_finite_o @ F3 )
             => ( ~ ( member_o @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_o @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4856_finite__induct,axiom,
    ! [F4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,F3: set_nat] :
              ( ( finite_finite_nat @ F3 )
             => ( ~ ( member_nat @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_nat @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4857_finite__induct,axiom,
    ! [F4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,F3: set_int] :
              ( ( finite_finite_int @ F3 )
             => ( ~ ( member_int @ X5 @ F3 )
               => ( ( P @ F3 )
                 => ( P @ ( insert_int @ X5 @ F3 ) ) ) ) )
         => ( P @ F4 ) ) ) ) ).

% finite_induct
thf(fact_4858_finite_Osimps,axiom,
    ( finite4343798906461161616at_nat
    = ( ^ [A6: set_Pr4329608150637261639at_nat] :
          ( ( A6 = bot_bo228742789529271731at_nat )
          | ? [A5: set_Pr4329608150637261639at_nat,B7: produc3843707927480180839at_nat] :
              ( ( A6
                = ( insert9069300056098147895at_nat @ B7 @ A5 ) )
              & ( finite4343798906461161616at_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4859_finite_Osimps,axiom,
    ( finite3207457112153483333omplex
    = ( ^ [A6: set_complex] :
          ( ( A6 = bot_bot_set_complex )
          | ? [A5: set_complex,B7: complex] :
              ( ( A6
                = ( insert_complex @ B7 @ A5 ) )
              & ( finite3207457112153483333omplex @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4860_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat] :
          ( ( A6 = bot_bo2099793752762293965at_nat )
          | ? [A5: set_Pr1261947904930325089at_nat,B7: product_prod_nat_nat] :
              ( ( A6
                = ( insert8211810215607154385at_nat @ B7 @ A5 ) )
              & ( finite6177210948735845034at_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4861_finite_Osimps,axiom,
    ( finite_finite_o
    = ( ^ [A6: set_o] :
          ( ( A6 = bot_bot_set_o )
          | ? [A5: set_o,B7: $o] :
              ( ( A6
                = ( insert_o @ B7 @ A5 ) )
              & ( finite_finite_o @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4862_finite_Osimps,axiom,
    ( finite_finite_nat
    = ( ^ [A6: set_nat] :
          ( ( A6 = bot_bot_set_nat )
          | ? [A5: set_nat,B7: nat] :
              ( ( A6
                = ( insert_nat @ B7 @ A5 ) )
              & ( finite_finite_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4863_finite_Osimps,axiom,
    ( finite_finite_int
    = ( ^ [A6: set_int] :
          ( ( A6 = bot_bot_set_int )
          | ? [A5: set_int,B7: int] :
              ( ( A6
                = ( insert_int @ B7 @ A5 ) )
              & ( finite_finite_int @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4864_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A8: set_Pr4329608150637261639at_nat] :
              ( ? [A3: produc3843707927480180839at_nat] :
                  ( A
                  = ( insert9069300056098147895at_nat @ A3 @ A8 ) )
             => ~ ( finite4343798906461161616at_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4865_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A8: set_complex] :
              ( ? [A3: complex] :
                  ( A
                  = ( insert_complex @ A3 @ A8 ) )
             => ~ ( finite3207457112153483333omplex @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4866_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A8: set_Pr1261947904930325089at_nat] :
              ( ? [A3: product_prod_nat_nat] :
                  ( A
                  = ( insert8211810215607154385at_nat @ A3 @ A8 ) )
             => ~ ( finite6177210948735845034at_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4867_finite_Ocases,axiom,
    ! [A: set_o] :
      ( ( finite_finite_o @ A )
     => ( ( A != bot_bot_set_o )
       => ~ ! [A8: set_o] :
              ( ? [A3: $o] :
                  ( A
                  = ( insert_o @ A3 @ A8 ) )
             => ~ ( finite_finite_o @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4868_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A8: set_nat] :
              ( ? [A3: nat] :
                  ( A
                  = ( insert_nat @ A3 @ A8 ) )
             => ~ ( finite_finite_nat @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4869_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A8: set_int] :
              ( ? [A3: int] :
                  ( A
                  = ( insert_int @ A3 @ A8 ) )
             => ~ ( finite_finite_int @ A8 ) ) ) ) ).

% finite.cases
thf(fact_4870_finite__subset__induct_H,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A3: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F3 )
               => ( ( member8757157785044589968at_nat @ A3 @ A4 )
                 => ( ( ord_le1268244103169919719at_nat @ F3 @ A4 )
                   => ( ~ ( member8757157785044589968at_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert9069300056098147895at_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4871_finite__subset__induct_H,axiom,
    ! [F4: set_real,A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( ord_less_eq_set_real @ F4 @ A4 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A3 @ A4 )
                 => ( ( ord_less_eq_set_real @ F3 @ A4 )
                   => ( ~ ( member_real @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4872_finite__subset__induct_H,axiom,
    ! [F4: set_complex,A4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( ord_le211207098394363844omplex @ F4 @ A4 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A3 @ A4 )
                 => ( ( ord_le211207098394363844omplex @ F3 @ A4 )
                   => ( ~ ( member_complex @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_complex @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4873_finite__subset__induct_H,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( member8440522571783428010at_nat @ A3 @ A4 )
                 => ( ( ord_le3146513528884898305at_nat @ F3 @ A4 )
                   => ( ~ ( member8440522571783428010at_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert8211810215607154385at_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4874_finite__subset__induct_H,axiom,
    ! [F4: set_o,A4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( ord_less_eq_set_o @ F4 @ A4 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A3: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ( ord_less_eq_set_o @ F3 @ A4 )
                   => ( ~ ( member_o @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_o @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4875_finite__subset__induct_H,axiom,
    ! [F4: set_int,A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A4 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ( ord_less_eq_set_int @ F3 @ A4 )
                   => ( ~ ( member_int @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_int @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4876_finite__subset__induct_H,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ( ord_less_eq_set_nat @ F3 @ A4 )
                   => ( ~ ( member_nat @ A3 @ F3 )
                     => ( ( P @ F3 )
                       => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4877_finite__subset__induct,axiom,
    ! [F4: set_Pr4329608150637261639at_nat,A4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F4 )
     => ( ( ord_le1268244103169919719at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A3: produc3843707927480180839at_nat,F3: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F3 )
               => ( ( member8757157785044589968at_nat @ A3 @ A4 )
                 => ( ~ ( member8757157785044589968at_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert9069300056098147895at_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4878_finite__subset__induct,axiom,
    ! [F4: set_real,A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F4 )
     => ( ( ord_less_eq_set_real @ F4 @ A4 )
       => ( ( P @ bot_bot_set_real )
         => ( ! [A3: real,F3: set_real] :
                ( ( finite_finite_real @ F3 )
               => ( ( member_real @ A3 @ A4 )
                 => ( ~ ( member_real @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_real @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4879_finite__subset__induct,axiom,
    ! [F4: set_complex,A4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F4 )
     => ( ( ord_le211207098394363844omplex @ F4 @ A4 )
       => ( ( P @ bot_bot_set_complex )
         => ( ! [A3: complex,F3: set_complex] :
                ( ( finite3207457112153483333omplex @ F3 )
               => ( ( member_complex @ A3 @ A4 )
                 => ( ~ ( member_complex @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_complex @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4880_finite__subset__induct,axiom,
    ! [F4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F4 )
     => ( ( ord_le3146513528884898305at_nat @ F4 @ A4 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A3: product_prod_nat_nat,F3: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F3 )
               => ( ( member8440522571783428010at_nat @ A3 @ A4 )
                 => ( ~ ( member8440522571783428010at_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert8211810215607154385at_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4881_finite__subset__induct,axiom,
    ! [F4: set_o,A4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ F4 )
     => ( ( ord_less_eq_set_o @ F4 @ A4 )
       => ( ( P @ bot_bot_set_o )
         => ( ! [A3: $o,F3: set_o] :
                ( ( finite_finite_o @ F3 )
               => ( ( member_o @ A3 @ A4 )
                 => ( ~ ( member_o @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_o @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4882_finite__subset__induct,axiom,
    ! [F4: set_int,A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F4 )
     => ( ( ord_less_eq_set_int @ F4 @ A4 )
       => ( ( P @ bot_bot_set_int )
         => ( ! [A3: int,F3: set_int] :
                ( ( finite_finite_int @ F3 )
               => ( ( member_int @ A3 @ A4 )
                 => ( ~ ( member_int @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_int @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4883_finite__subset__induct,axiom,
    ! [F4: set_nat,A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F4 )
     => ( ( ord_less_eq_set_nat @ F4 @ A4 )
       => ( ( P @ bot_bot_set_nat )
         => ( ! [A3: nat,F3: set_nat] :
                ( ( finite_finite_nat @ F3 )
               => ( ( member_nat @ A3 @ A4 )
                 => ( ~ ( member_nat @ A3 @ F3 )
                   => ( ( P @ F3 )
                     => ( P @ ( insert_nat @ A3 @ F3 ) ) ) ) ) )
           => ( P @ F4 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4884_finite__empty__induct,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: produc3843707927480180839at_nat,A8: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A8 )
             => ( ( member8757157785044589968at_nat @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ A3 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
         => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_4885_finite__empty__induct,axiom,
    ! [A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ( member_real @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_real @ A8 @ ( insert_real @ A3 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_4886_finite__empty__induct,axiom,
    ! [A4: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: complex,A8: set_complex] :
              ( ( finite3207457112153483333omplex @ A8 )
             => ( ( member_complex @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ A3 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_4887_finite__empty__induct,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: product_prod_nat_nat,A8: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A8 )
             => ( ( member8440522571783428010at_nat @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ A3 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_4888_finite__empty__induct,axiom,
    ! [A4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ( member_o @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_o @ A8 @ ( insert_o @ A3 @ bot_bot_set_o ) ) ) ) ) )
         => ( P @ bot_bot_set_o ) ) ) ) ).

% finite_empty_induct
thf(fact_4889_finite__empty__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ( member_int @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_int @ A8 @ ( insert_int @ A3 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_4890_finite__empty__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ A4 )
       => ( ! [A3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ( member_nat @ A3 @ A8 )
               => ( ( P @ A8 )
                 => ( P @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ A3 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_4891_infinite__coinduct,axiom,
    ! [X6: set_Pr4329608150637261639at_nat > $o,A4: set_Pr4329608150637261639at_nat] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_Pr4329608150637261639at_nat] :
            ( ( X6 @ A8 )
           => ? [X: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X @ A8 )
                & ( ( X6 @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) )
                  | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A8 @ ( insert9069300056098147895at_nat @ X @ bot_bo228742789529271731at_nat ) ) ) ) ) )
       => ~ ( finite4343798906461161616at_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4892_infinite__coinduct,axiom,
    ! [X6: set_complex > $o,A4: set_complex] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_complex] :
            ( ( X6 @ A8 )
           => ? [X: complex] :
                ( ( member_complex @ X @ A8 )
                & ( ( X6 @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A8 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4893_infinite__coinduct,axiom,
    ! [X6: set_Pr1261947904930325089at_nat > $o,A4: set_Pr1261947904930325089at_nat] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_Pr1261947904930325089at_nat] :
            ( ( X6 @ A8 )
           => ? [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ A8 )
                & ( ( X6 @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A8 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4894_infinite__coinduct,axiom,
    ! [X6: set_o > $o,A4: set_o] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_o] :
            ( ( X6 @ A8 )
           => ? [X: $o] :
                ( ( member_o @ X @ A8 )
                & ( ( X6 @ ( minus_minus_set_o @ A8 @ ( insert_o @ X @ bot_bot_set_o ) ) )
                  | ~ ( finite_finite_o @ ( minus_minus_set_o @ A8 @ ( insert_o @ X @ bot_bot_set_o ) ) ) ) ) )
       => ~ ( finite_finite_o @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4895_infinite__coinduct,axiom,
    ! [X6: set_int > $o,A4: set_int] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_int] :
            ( ( X6 @ A8 )
           => ? [X: int] :
                ( ( member_int @ X @ A8 )
                & ( ( X6 @ ( minus_minus_set_int @ A8 @ ( insert_int @ X @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A8 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4896_infinite__coinduct,axiom,
    ! [X6: set_nat > $o,A4: set_nat] :
      ( ( X6 @ A4 )
     => ( ! [A8: set_nat] :
            ( ( X6 @ A8 )
           => ? [X: nat] :
                ( ( member_nat @ X @ A8 )
                & ( ( X6 @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A8 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A4 ) ) ) ).

% infinite_coinduct
thf(fact_4897_infinite__remove,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S3 )
     => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% infinite_remove
thf(fact_4898_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_4899_infinite__remove,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S3 )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_4900_infinite__remove,axiom,
    ! [S3: set_o,A: $o] :
      ( ~ ( finite_finite_o @ S3 )
     => ~ ( finite_finite_o @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ).

% infinite_remove
thf(fact_4901_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_4902_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_4903_finite__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_4904_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_4905_finite__linorder__min__induct,axiom,
    ! [A4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X: $o] :
                    ( ( member_o @ X @ A8 )
                   => ( ord_less_o @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4906_finite__linorder__min__induct,axiom,
    ! [A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X: real] :
                    ( ( member_real @ X @ A8 )
                   => ( ord_less_real @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4907_finite__linorder__min__induct,axiom,
    ! [A4: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X: rat] :
                    ( ( member_rat @ X @ A8 )
                   => ( ord_less_rat @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4908_finite__linorder__min__induct,axiom,
    ! [A4: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X: num] :
                    ( ( member_num @ X @ A8 )
                   => ( ord_less_num @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4909_finite__linorder__min__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X: nat] :
                    ( ( member_nat @ X @ A8 )
                   => ( ord_less_nat @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4910_finite__linorder__min__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X: int] :
                    ( ( member_int @ X @ A8 )
                   => ( ord_less_int @ B3 @ X ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4911_finite__linorder__max__induct,axiom,
    ! [A4: set_o,P: set_o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [B3: $o,A8: set_o] :
              ( ( finite_finite_o @ A8 )
             => ( ! [X: $o] :
                    ( ( member_o @ X @ A8 )
                   => ( ord_less_o @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_o @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4912_finite__linorder__max__induct,axiom,
    ! [A4: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B3: real,A8: set_real] :
              ( ( finite_finite_real @ A8 )
             => ( ! [X: real] :
                    ( ( member_real @ X @ A8 )
                   => ( ord_less_real @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_real @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4913_finite__linorder__max__induct,axiom,
    ! [A4: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A4 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B3: rat,A8: set_rat] :
              ( ( finite_finite_rat @ A8 )
             => ( ! [X: rat] :
                    ( ( member_rat @ X @ A8 )
                   => ( ord_less_rat @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_rat @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4914_finite__linorder__max__induct,axiom,
    ! [A4: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A4 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B3: num,A8: set_num] :
              ( ( finite_finite_num @ A8 )
             => ( ! [X: num] :
                    ( ( member_num @ X @ A8 )
                   => ( ord_less_num @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_num @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4915_finite__linorder__max__induct,axiom,
    ! [A4: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B3: nat,A8: set_nat] :
              ( ( finite_finite_nat @ A8 )
             => ( ! [X: nat] :
                    ( ( member_nat @ X @ A8 )
                   => ( ord_less_nat @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_nat @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4916_finite__linorder__max__induct,axiom,
    ! [A4: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B3: int,A8: set_int] :
              ( ( finite_finite_int @ A8 )
             => ( ! [X: int] :
                    ( ( member_int @ X @ A8 )
                   => ( ord_less_int @ X @ B3 ) )
               => ( ( P @ A8 )
                 => ( P @ ( insert_int @ B3 @ A8 ) ) ) ) )
         => ( P @ A4 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4917_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4918_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4919_finite__ranking__induct,axiom,
    ! [S3: set_o,P: set_o > $o,F: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y6: $o] :
                    ( ( member_o @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4920_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4921_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4922_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X5: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_real @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4923_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X5: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_complex @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4924_finite__ranking__induct,axiom,
    ! [S3: set_o,P: set_o > $o,F: $o > num] :
      ( ( finite_finite_o @ S3 )
     => ( ( P @ bot_bot_set_o )
       => ( ! [X5: $o,S4: set_o] :
              ( ( finite_finite_o @ S4 )
             => ( ! [Y6: $o] :
                    ( ( member_o @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_o @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4925_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X5: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_nat @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4926_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X5: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X5 ) ) )
               => ( ( P @ S4 )
                 => ( P @ ( insert_int @ X5 @ S4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4927_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > real,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4928_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X3: $o > real,Y: $o > real] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4929_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > real,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4930_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > real,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4931_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > real,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4932_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > rat,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I4: real] :
                ( ( member_real @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4933_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_o,X3: $o > rat,Y: $o > rat] :
      ( ( finite_finite_o
        @ ( collect_o
          @ ^ [I4: $o] :
              ( ( member_o @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_o
          @ ( collect_o
            @ ^ [I4: $o] :
                ( ( member_o @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4934_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > rat,Y: int > rat] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I4: int] :
              ( ( member_int @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I4: int] :
                ( ( member_int @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4935_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > rat,Y: 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 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I4: nat] :
                ( ( member_nat @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4936_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > rat,Y: complex > rat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
              & ( ( X3 @ I4 )
               != zero_zero_rat ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( Y @ I4 )
                 != zero_zero_rat ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I4: complex] :
                ( ( member_complex @ I4 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I4 ) @ ( Y @ I4 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_4937_concat__bit__Suc,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_concat_bit @ ( suc @ N ) @ K2 @ L )
      = ( plus_plus_int @ ( modulo_modulo_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).

% concat_bit_Suc
thf(fact_4938_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_4939_concat__bit__assoc,axiom,
    ! [N: nat,K2: int,M2: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N @ K2 @ ( bit_concat_bit @ M2 @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M2 @ N ) @ ( bit_concat_bit @ N @ K2 @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_4940_set__encode__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( nat_set_encode @ A4 )
            = ( nat_set_encode @ B4 ) )
          = ( A4 = B4 ) ) ) ) ).

% set_encode_eq
thf(fact_4941_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( nat_set_encode @ A4 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_4942_infinite__growing,axiom,
    ! [X6: set_o] :
      ( ( X6 != bot_bot_set_o )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ X6 )
           => ? [Xa: $o] :
                ( ( member_o @ Xa @ X6 )
                & ( ord_less_o @ X5 @ Xa ) ) )
       => ~ ( finite_finite_o @ X6 ) ) ) ).

% infinite_growing
thf(fact_4943_infinite__growing,axiom,
    ! [X6: set_real] :
      ( ( X6 != bot_bot_set_real )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ X6 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X6 )
                & ( ord_less_real @ X5 @ Xa ) ) )
       => ~ ( finite_finite_real @ X6 ) ) ) ).

% infinite_growing
thf(fact_4944_infinite__growing,axiom,
    ! [X6: set_rat] :
      ( ( X6 != bot_bot_set_rat )
     => ( ! [X5: rat] :
            ( ( member_rat @ X5 @ X6 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X6 )
                & ( ord_less_rat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X6 ) ) ) ).

% infinite_growing
thf(fact_4945_infinite__growing,axiom,
    ! [X6: set_num] :
      ( ( X6 != bot_bot_set_num )
     => ( ! [X5: num] :
            ( ( member_num @ X5 @ X6 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X6 )
                & ( ord_less_num @ X5 @ Xa ) ) )
       => ~ ( finite_finite_num @ X6 ) ) ) ).

% infinite_growing
thf(fact_4946_infinite__growing,axiom,
    ! [X6: set_nat] :
      ( ( X6 != bot_bot_set_nat )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ X6 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X6 )
                & ( ord_less_nat @ X5 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X6 ) ) ) ).

% infinite_growing
thf(fact_4947_infinite__growing,axiom,
    ! [X6: set_int] :
      ( ( X6 != bot_bot_set_int )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ X6 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X6 )
                & ( ord_less_int @ X5 @ Xa ) ) )
       => ~ ( finite_finite_int @ X6 ) ) ) ).

% infinite_growing
thf(fact_4948_ex__min__if__finite,axiom,
    ! [S3: set_o] :
      ( ( finite_finite_o @ S3 )
     => ( ( S3 != bot_bot_set_o )
       => ? [X5: $o] :
            ( ( member_o @ X5 @ S3 )
            & ~ ? [Xa: $o] :
                  ( ( member_o @ Xa @ S3 )
                  & ( ord_less_o @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4949_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X5: real] :
            ( ( member_real @ X5 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4950_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X5: rat] :
            ( ( member_rat @ X5 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4951_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X5: num] :
            ( ( member_num @ X5 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4952_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X5: nat] :
            ( ( member_nat @ X5 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4953_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X5: int] :
            ( ( member_int @ X5 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X5 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_4954_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4955_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4956_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4957_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_4958_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_4959_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_4960_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_4961_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_4962_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_4963_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_4964_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_4965_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_4966_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_4967_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_4968_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_4969_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_4970_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4971_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4972_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4973_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_4974_nat__dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ one_one_nat )
      = ( M2 = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_4975_dvd__1__left,axiom,
    ! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).

% dvd_1_left
thf(fact_4976_dvd__1__iff__1,axiom,
    ! [M2: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( M2
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_4977_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4978_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4979_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4980_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4981_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_4982_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4983_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4984_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4985_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4986_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_4987_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_4988_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4989_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4990_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4991_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4992_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_4993_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4994_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4995_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4996_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4997_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_4998_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_4999_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_5000_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_5001_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_5002_signed__take__bit__numeral__of__1,axiom,
    ! [K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K2 ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_5003_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_5004_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_5005_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_5006_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_5007_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_5008_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
        = A4 ) ) ).

% set_encode_inverse
thf(fact_5009_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_5010_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_5011_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5012_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5013_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_5014_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_5015_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_5016_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_5017_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_5018_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_5019_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_5020_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_5021_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_5022_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_5023_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_5024_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_5025_signed__take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_bit0
thf(fact_5026_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_5027_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_5028_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_5029_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_5030_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_5031_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_5032_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_5033_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_5034_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_5035_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_5036_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_5037_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_5038_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_5039_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_5040_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_5041_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_5042_even__diff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ).

% even_diff_nat
thf(fact_5043_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_5044_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_5045_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_5046_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_5047_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_5048_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_5049_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_5050_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_5051_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_5052_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_5053_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_5054_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_5055_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_5056_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_5057_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_5058_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_5059_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_5060_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_5061_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_5062_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_5063_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_5064_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_5065_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_5066_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_5067_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_5068_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5069_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5070_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5071_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5072_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_5073_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5074_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5075_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5076_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5077_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5078_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_5079_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_5080_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_5081_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_5082_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_5083_dvd__power__same,axiom,
    ! [X3: code_integer,Y: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5084_dvd__power__same,axiom,
    ! [X3: nat,Y: nat,N: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5085_dvd__power__same,axiom,
    ! [X3: real,Y: real,N: nat] :
      ( ( dvd_dvd_real @ X3 @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5086_dvd__power__same,axiom,
    ! [X3: int,Y: int,N: nat] :
      ( ( dvd_dvd_int @ X3 @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5087_dvd__power__same,axiom,
    ! [X3: complex,Y: complex,N: nat] :
      ( ( dvd_dvd_complex @ X3 @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ N ) ) ) ).

% dvd_power_same
thf(fact_5088_dvd__diff__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ M2 )
     => ( ( dvd_dvd_nat @ K2 @ N )
       => ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% dvd_diff_nat
thf(fact_5089_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ A ) )
        @ ( collect_complex
          @ ^ [C4: complex] : ( dvd_dvd_complex @ C4 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5090_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ A ) )
        @ ( collect_int
          @ ^ [C4: int] : ( dvd_dvd_int @ C4 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5091_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C4: code_integer] : ( dvd_dvd_Code_integer @ C4 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5092_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ A ) )
        @ ( collect_nat
          @ ^ [C4: nat] : ( dvd_dvd_nat @ C4 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5093_even__signed__take__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M2 @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_5094_even__signed__take__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M2 @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_5095_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_5096_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5097_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5098_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_5099_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5100_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5101_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_5102_div__power,axiom,
    ! [B: code_integer,A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% div_power
thf(fact_5103_div__power,axiom,
    ! [B: nat,A: nat,N: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).

% div_power
thf(fact_5104_div__power,axiom,
    ! [B: int,A: int,N: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N )
        = ( divide_divide_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% div_power
thf(fact_5105_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5106_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5107_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ ( power_power_real @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5108_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5109_le__imp__power__dvd,axiom,
    ! [M2: nat,N: nat,A: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ ( power_power_complex @ A @ N ) ) ) ).

% le_imp_power_dvd
thf(fact_5110_power__le__dvd,axiom,
    ! [A: code_integer,N: nat,B: code_integer,M2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5111_power__le__dvd,axiom,
    ! [A: nat,N: nat,B: nat,M2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5112_power__le__dvd,axiom,
    ! [A: real,N: nat,B: real,M2: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5113_power__le__dvd,axiom,
    ! [A: int,N: nat,B: int,M2: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5114_power__le__dvd,axiom,
    ! [A: complex,N: nat,B: complex,M2: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N ) @ B )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M2 ) @ B ) ) ) ).

% power_le_dvd
thf(fact_5115_dvd__power__le,axiom,
    ! [X3: code_integer,Y: code_integer,N: nat,M2: nat] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N ) @ ( power_8256067586552552935nteger @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5116_dvd__power__le,axiom,
    ! [X3: nat,Y: nat,N: nat,M2: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5117_dvd__power__le,axiom,
    ! [X3: real,Y: real,N: nat,M2: nat] :
      ( ( dvd_dvd_real @ X3 @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5118_dvd__power__le,axiom,
    ! [X3: int,Y: int,N: nat,M2: nat] :
      ( ( dvd_dvd_int @ X3 @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5119_dvd__power__le,axiom,
    ! [X3: complex,Y: complex,N: nat,M2: nat] :
      ( ( dvd_dvd_complex @ X3 @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5120_nat__dvd__not__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% nat_dvd_not_less
thf(fact_5121_dvd__minus__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_minus_self
thf(fact_5122_dvd__diffD,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K2 @ N )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K2 @ M2 ) ) ) ) ).

% dvd_diffD
thf(fact_5123_dvd__diffD1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ ( minus_minus_nat @ M2 @ N ) )
     => ( ( dvd_dvd_nat @ K2 @ M2 )
       => ( ( ord_less_eq_nat @ N @ M2 )
         => ( dvd_dvd_nat @ K2 @ N ) ) ) ) ).

% dvd_diffD1
thf(fact_5124_less__eq__dvd__minus,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
        = ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_5125_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_5126_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_5127_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_5128_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5129_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5130_even__numeral,axiom,
    ! [N: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) ).

% even_numeral
thf(fact_5131_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_5132_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_5133_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_5134_dvd__imp__le,axiom,
    ! [K2: nat,N: nat] :
      ( ( dvd_dvd_nat @ K2 @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat @ K2 @ N ) ) ) ).

% dvd_imp_le
thf(fact_5135_dvd__mult__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% dvd_mult_cancel
thf(fact_5136_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X4: nat] :
          ( collect_nat
          @ ^ [N3: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_5137_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( ( modulo_modulo_nat @ M2 @ Q3 )
          = ( modulo_modulo_nat @ N @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_5138_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_5139_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_5140_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_5141_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_5142_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_5143_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_5144_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5145_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5146_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_5147_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_5148_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_5149_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_5150_bit__eq__rec,axiom,
    ( ( ^ [Y5: code_integer,Z: code_integer] : ( Y5 = Z ) )
    = ( ^ [A6: code_integer,B7: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B7 ) )
          & ( ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B7 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5151_bit__eq__rec,axiom,
    ( ( ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    = ( ^ [A6: nat,B7: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B7 ) )
          & ( ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B7 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5152_bit__eq__rec,axiom,
    ( ( ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    = ( ^ [A6: int,B7: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B7 ) )
          & ( ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B7 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_5153_subset__decode__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% subset_decode_imp_le
thf(fact_5154_dvd__power__iff,axiom,
    ! [X3: code_integer,M2: nat,N: nat] :
      ( ( X3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M2 ) @ ( power_8256067586552552935nteger @ X3 @ N ) )
        = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5155_dvd__power__iff,axiom,
    ! [X3: nat,M2: nat,N: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M2 ) @ ( power_power_nat @ X3 @ N ) )
        = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5156_dvd__power__iff,axiom,
    ! [X3: int,M2: nat,N: nat] :
      ( ( X3 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M2 ) @ ( power_power_int @ X3 @ N ) )
        = ( ( dvd_dvd_int @ X3 @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5157_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_5158_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_5159_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_5160_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_5161_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_5162_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_5163_div2__even__ext__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X3 = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_5164_dvd__mult__cancel1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M2 @ N ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_5165_dvd__mult__cancel2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N @ M2 ) @ M2 )
        = ( N = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_5166_dvd__minus__add,axiom,
    ! [Q3: nat,N: nat,R2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R2 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q3 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M2 ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_5167_power__dvd__imp__le,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_5168_mod__nat__eqI,axiom,
    ! [R2: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ R2 @ N )
     => ( ( ord_less_eq_nat @ R2 @ M2 )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R2 ) )
         => ( ( modulo_modulo_nat @ M2 @ N )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_5169_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_5170_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_5171_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_5172_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_5173_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_5174_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_5175_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_5176_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_5177_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_5178_power__mono__odd,axiom,
    ! [N: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_5179_power__mono__odd,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_5180_power__mono__odd,axiom,
    ! [N: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_odd
thf(fact_5181_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_5182_dvd__power__iff__le,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K2 @ M2 ) @ ( power_power_nat @ K2 @ N ) )
        = ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% dvd_power_iff_le
thf(fact_5183_signed__take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% signed_take_bit_int_less_exp
thf(fact_5184_even__unset__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5185_even__unset__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5186_even__unset__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M2 = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_5187_even__set__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5188_even__set__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5189_even__set__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M2 != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_5190_even__flip__bit__iff,axiom,
    ! [M2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M2 @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5191_even__flip__bit__iff,axiom,
    ! [M2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5192_even__flip__bit__iff,axiom,
    ! [M2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M2 = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_5193_even__diff__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K2 @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_diff_iff
thf(fact_5194_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_5195_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_5196_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_5197_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_5198_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_5199_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_5200_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_5201_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_5202_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_5203_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_5204_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_5205_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_5206_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_5207_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_5208_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_5209_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_5210_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_5211_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_5212_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_5213_signed__take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_5214_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_5215_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_5216_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_5217_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_5218_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_5219_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5220_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5221_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5222_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_5223_crossproduct__eq,axiom,
    ! [W: real,Y: real,X3: real,Z2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X3 @ Z2 ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z2 ) @ ( times_times_real @ X3 @ Y ) ) )
      = ( ( W = X3 )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_5224_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X3: rat,Z2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X3 @ Z2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z2 ) @ ( times_times_rat @ X3 @ Y ) ) )
      = ( ( W = X3 )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_5225_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X3: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X3 @ Z2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X3 @ Y ) ) )
      = ( ( W = X3 )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_5226_crossproduct__eq,axiom,
    ! [W: int,Y: int,X3: int,Z2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X3 @ Z2 ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X3 @ Y ) ) )
      = ( ( W = X3 )
        | ( Y = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_5227_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_5228_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_5229_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_5230_signed__take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
     => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) ) ) ).

% signed_take_bit_int_less_eq
thf(fact_5231_even__mask__div__iff_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_5232_even__mask__div__iff_H,axiom,
    ! [M2: 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 ) ) @ M2 ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_5233_even__mask__div__iff_H,axiom,
    ! [M2: 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 ) ) @ M2 ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% even_mask_div_iff'
thf(fact_5234_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_5235_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_5236_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_5237_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_5238_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_5239_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_5240_even__mask__div__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_5241_even__mask__div__iff,axiom,
    ! [M2: 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 ) ) @ M2 ) @ 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 @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_5242_even__mask__div__iff,axiom,
    ! [M2: 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 ) ) @ M2 ) @ 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 @ M2 @ N ) ) ) ).

% even_mask_div_iff
thf(fact_5243_set__decode__plus__power__2,axiom,
    ! [N: nat,Z2: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z2 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z2 ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_5244_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5245_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M2: 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 ) ) @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M2 @ 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 @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5246_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M2: 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 ) ) @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
      = ( ( ord_less_nat @ N @ M2 )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M2 @ 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 @ M2 ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_5247_pow__divides__pow__iff,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_5248_pow__divides__pow__iff,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_5249_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_5250_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_5251_inf__period_I4_J,axiom,
    ! [D: code_integer,D2: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D2 )
     => ! [X: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X @ ( times_3573771949741848930nteger @ K4 @ D2 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5252_inf__period_I4_J,axiom,
    ! [D: real,D2: real,T: real] :
      ( ( dvd_dvd_real @ D @ D2 )
     => ! [X: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X @ ( times_times_real @ K4 @ D2 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5253_inf__period_I4_J,axiom,
    ! [D: rat,D2: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D2 )
     => ! [X: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ K4 @ D2 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5254_inf__period_I4_J,axiom,
    ! [D: int,D2: int,T: int] :
      ( ( dvd_dvd_int @ D @ D2 )
     => ! [X: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5255_inf__period_I3_J,axiom,
    ! [D: code_integer,D2: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D2 )
     => ! [X: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X @ ( times_3573771949741848930nteger @ K4 @ D2 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5256_inf__period_I3_J,axiom,
    ! [D: real,D2: real,T: real] :
      ( ( dvd_dvd_real @ D @ D2 )
     => ! [X: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X @ ( times_times_real @ K4 @ D2 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5257_inf__period_I3_J,axiom,
    ! [D: rat,D2: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D2 )
     => ! [X: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ K4 @ D2 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5258_inf__period_I3_J,axiom,
    ! [D: int,D2: int,T: int] :
      ( ( dvd_dvd_int @ D @ D2 )
     => ! [X: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X @ ( times_times_int @ K4 @ D2 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5259_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X4: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X4 ) ) )
      = ( ? [X4: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X4 @ zero_z3403309356797280102nteger ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5260_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X4: real] : ( P @ ( times_times_real @ L @ X4 ) ) )
      = ( ? [X4: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5261_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X4: rat] : ( P @ ( times_times_rat @ L @ X4 ) ) )
      = ( ? [X4: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5262_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5263_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X4: int] : ( P @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5264_vebt__buildup_Oelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X3
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va3: nat] :
                ( ( X3
                  = ( suc @ ( suc @ Va3 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_5265_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_5266_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_5267_length__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5268_length__replicate,axiom,
    ! [N: nat,X3: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5269_length__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5270_length__replicate,axiom,
    ! [N: nat,X3: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X3 ) )
      = N ) ).

% length_replicate
thf(fact_5271_in__set__replicate,axiom,
    ! [X3: complex,N: nat,Y: complex] :
      ( ( member_complex @ X3 @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5272_in__set__replicate,axiom,
    ! [X3: real,N: nat,Y: real] :
      ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5273_in__set__replicate,axiom,
    ! [X3: $o,N: nat,Y: $o] :
      ( ( member_o @ X3 @ ( set_o2 @ ( replicate_o @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5274_in__set__replicate,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5275_in__set__replicate,axiom,
    ! [X3: nat,N: nat,Y: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5276_in__set__replicate,axiom,
    ! [X3: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X3 = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_5277_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5278_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P @ X4 ) ) )
      = ( ( P @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_5279_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5280_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P @ X4 ) ) )
      = ( ( P @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_5281_set__replicate,axiom,
    ! [N: nat,X3: produc3843707927480180839at_nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X3 ) )
        = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% set_replicate
thf(fact_5282_set__replicate,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X3 ) )
        = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_5283_set__replicate,axiom,
    ! [N: nat,X3: product_prod_nat_nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) )
        = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_replicate
thf(fact_5284_set__replicate,axiom,
    ! [N: nat,X3: $o] :
      ( ( N != zero_zero_nat )
     => ( ( set_o2 @ ( replicate_o @ N @ X3 ) )
        = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% set_replicate
thf(fact_5285_set__replicate,axiom,
    ! [N: nat,X3: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X3 ) )
        = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_5286_set__replicate,axiom,
    ! [N: nat,X3: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X3 ) )
        = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_5287_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% dvd_antisym
thf(fact_5288_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_5289_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_5290_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_5291_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_5292_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_5293_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_5294_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_5295_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_5296_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_5297_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_5298_set__replicate__Suc,axiom,
    ! [N: nat,X3: produc3843707927480180839at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N ) @ X3 ) )
      = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ).

% set_replicate_Suc
thf(fact_5299_set__replicate__Suc,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X3 ) )
      = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_5300_set__replicate__Suc,axiom,
    ! [N: nat,X3: product_prod_nat_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N ) @ X3 ) )
      = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).

% set_replicate_Suc
thf(fact_5301_set__replicate__Suc,axiom,
    ! [N: nat,X3: $o] :
      ( ( set_o2 @ ( replicate_o @ ( suc @ N ) @ X3 ) )
      = ( insert_o @ X3 @ bot_bot_set_o ) ) ).

% set_replicate_Suc
thf(fact_5302_set__replicate__Suc,axiom,
    ! [N: nat,X3: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X3 ) )
      = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_5303_set__replicate__Suc,axiom,
    ! [N: nat,X3: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X3 ) )
      = ( insert_int @ X3 @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_5304_set__replicate__conv__if,axiom,
    ! [N: nat,X3: produc3843707927480180839at_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X3 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N @ X3 ) )
          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5305_set__replicate__conv__if,axiom,
    ! [N: nat,X3: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X3 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X3 ) )
          = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5306_set__replicate__conv__if,axiom,
    ! [N: nat,X3: product_prod_nat_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N @ X3 ) )
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5307_set__replicate__conv__if,axiom,
    ! [N: nat,X3: $o] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X3 ) )
          = bot_bot_set_o ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_o2 @ ( replicate_o @ N @ X3 ) )
          = ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5308_set__replicate__conv__if,axiom,
    ! [N: nat,X3: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X3 ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X3 ) )
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5309_set__replicate__conv__if,axiom,
    ! [N: nat,X3: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X3 ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X3 ) )
          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_5310_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ X @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X ) ) ).

% minf(8)
thf(fact_5311_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ X @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X ) ) ).

% minf(8)
thf(fact_5312_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( ( ord_less_num @ X @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X ) ) ).

% minf(8)
thf(fact_5313_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X ) ) ).

% minf(8)
thf(fact_5314_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ X @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X ) ) ).

% minf(8)
thf(fact_5315_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ X @ Z3 )
     => ( ord_less_eq_real @ X @ T ) ) ).

% minf(6)
thf(fact_5316_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ X @ Z3 )
     => ( ord_less_eq_rat @ X @ T ) ) ).

% minf(6)
thf(fact_5317_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( ( ord_less_num @ X @ Z3 )
     => ( ord_less_eq_num @ X @ T ) ) ).

% minf(6)
thf(fact_5318_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ( ord_less_eq_nat @ X @ T ) ) ).

% minf(6)
thf(fact_5319_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ X @ Z3 )
     => ( ord_less_eq_int @ X @ T ) ) ).

% minf(6)
thf(fact_5320_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ord_less_eq_real @ T @ X ) ) ).

% pinf(8)
thf(fact_5321_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ( ord_less_eq_rat @ T @ X ) ) ).

% pinf(8)
thf(fact_5322_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( ( ord_less_num @ Z3 @ X )
     => ( ord_less_eq_num @ T @ X ) ) ).

% pinf(8)
thf(fact_5323_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ( ord_less_eq_nat @ T @ X ) ) ).

% pinf(8)
thf(fact_5324_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ Z3 @ X )
     => ( ord_less_eq_int @ T @ X ) ) ).

% pinf(8)
thf(fact_5325_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ~ ( ord_less_eq_real @ X @ T ) ) ).

% pinf(6)
thf(fact_5326_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ~ ( ord_less_eq_rat @ X @ T ) ) ).

% pinf(6)
thf(fact_5327_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X: num] :
      ( ( ord_less_num @ Z3 @ X )
     => ~ ( ord_less_eq_num @ X @ T ) ) ).

% pinf(6)
thf(fact_5328_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ~ ( ord_less_eq_nat @ X @ T ) ) ).

% pinf(6)
thf(fact_5329_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ Z3 @ X )
     => ~ ( ord_less_eq_int @ X @ T ) ) ).

% pinf(6)
thf(fact_5330_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ) ).

% Euclid_induct
thf(fact_5331_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X3: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X5: nat,Y4: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y4 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_5332_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X5: nat,Y4: nat] :
      ( ( dvd_dvd_nat @ D4 @ A )
      & ( dvd_dvd_nat @ D4 @ B )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) )
        | ( ( times_times_nat @ B @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y4 ) @ D4 ) ) ) ) ).

% bezout_add_nat
thf(fact_5333_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_5334_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5335_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ X @ Z3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5336_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ X @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5337_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5338_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ X @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) ) ) ) ).

% minf(10)
thf(fact_5339_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) ) ) ).

% minf(9)
thf(fact_5340_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ X @ Z3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) ) ) ).

% minf(9)
thf(fact_5341_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ X @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) ) ) ).

% minf(9)
thf(fact_5342_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) ) ) ).

% minf(9)
thf(fact_5343_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ X @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) ) ) ).

% minf(9)
thf(fact_5344_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5345_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5346_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5347_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5348_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ Z3 @ X )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_5349_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X @ S ) ) ) ) ).

% pinf(9)
thf(fact_5350_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X @ S ) ) ) ) ).

% pinf(9)
thf(fact_5351_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X @ S ) ) ) ) ).

% pinf(9)
thf(fact_5352_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X: nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X @ S ) ) ) ) ).

% pinf(9)
thf(fact_5353_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X: int] :
      ( ( ord_less_int @ Z3 @ X )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X @ S ) ) ) ) ).

% pinf(9)
thf(fact_5354_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X5: nat,Y4: nat] :
          ( ( dvd_dvd_nat @ D4 @ A )
          & ( dvd_dvd_nat @ D4 @ B )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ D4 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_5355_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N3: nat,A6: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5356_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,A6: int] : ( if_int @ ( N3 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_5357_vebt__buildup_Opelims,axiom,
    ! [X3: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X3
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va3: nat] :
                  ( ( X3
                    = ( suc @ ( suc @ Va3 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va3 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_5358_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_5359_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_5360_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_5361_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_5362_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_5363_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_5364_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_5365_signed__take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_bit1
thf(fact_5366_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A6: code_integer] : ( if_Code_integer @ ( N3 = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_5367_take__bit__rec,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A6: nat] : ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_5368_take__bit__rec,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A6: int] : ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_5369_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5370_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5371_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5372_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5373_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_5374_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5375_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5376_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5377_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5378_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_5379_Compl__subset__Compl__iff,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B4 ) )
      = ( ord_less_eq_set_nat @ B4 @ A4 ) ) ).

% Compl_subset_Compl_iff
thf(fact_5380_Compl__anti__mono,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ B4 ) @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).

% Compl_anti_mono
thf(fact_5381_compl__le__compl__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X3 ) ) ).

% compl_le_compl_iff
thf(fact_5382_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_5383_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_5384_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_5385_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_5386_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_5387_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_5388_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_5389_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_5390_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_5391_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_5392_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_5393_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_5394_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_5395_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_5396_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_5397_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_5398_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_5399_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_5400_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_5401_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_5402_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_5403_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_5404_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_5405_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_5406_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_5407_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_5408_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_5409_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_5410_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_5411_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_5412_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_5413_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5414_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5415_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5416_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5417_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_5418_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_5419_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_5420_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_5421_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_5422_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_5423_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5424_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5425_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5426_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5427_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5428_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5429_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5430_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5431_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5432_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5433_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5434_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5435_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5436_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5437_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5438_semiring__norm_I88_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit0 @ M2 )
     != ( bit1 @ N ) ) ).

% semiring_norm(88)
thf(fact_5439_semiring__norm_I89_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit1 @ M2 )
     != ( bit0 @ N ) ) ).

% semiring_norm(89)
thf(fact_5440_semiring__norm_I84_J,axiom,
    ! [N: num] :
      ( one
     != ( bit1 @ N ) ) ).

% semiring_norm(84)
thf(fact_5441_semiring__norm_I86_J,axiom,
    ! [M2: num] :
      ( ( bit1 @ M2 )
     != one ) ).

% semiring_norm(86)
thf(fact_5442_Compl__disjoint2,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ A4 )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint2
thf(fact_5443_Compl__disjoint2,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ A4 ) @ A4 )
      = bot_bot_set_o ) ).

% Compl_disjoint2
thf(fact_5444_Compl__disjoint2,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ A4 )
      = bot_bot_set_nat ) ).

% Compl_disjoint2
thf(fact_5445_Compl__disjoint2,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ A4 )
      = bot_bot_set_int ) ).

% Compl_disjoint2
thf(fact_5446_Compl__disjoint,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
      = bot_bo2099793752762293965at_nat ) ).

% Compl_disjoint
thf(fact_5447_Compl__disjoint,axiom,
    ! [A4: set_o] :
      ( ( inf_inf_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
      = bot_bot_set_o ) ).

% Compl_disjoint
thf(fact_5448_Compl__disjoint,axiom,
    ! [A4: set_nat] :
      ( ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ A4 ) )
      = bot_bot_set_nat ) ).

% Compl_disjoint
thf(fact_5449_Compl__disjoint,axiom,
    ! [A4: set_int] :
      ( ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ A4 ) )
      = bot_bot_set_int ) ).

% Compl_disjoint
thf(fact_5450_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_5451_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_5452_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_5453_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_5454_Diff__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A4 @ ( uminus6524753893492686040at_nat @ B4 ) )
      = ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ).

% Diff_Compl
thf(fact_5455_Diff__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( minus_minus_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B4 ) )
      = ( inf_inf_set_nat @ A4 @ B4 ) ) ).

% Diff_Compl
thf(fact_5456_Compl__Diff__eq,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( minus_3314409938677909166at_nat @ A4 @ B4 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ B4 ) ) ).

% Compl_Diff_eq
thf(fact_5457_Compl__Diff__eq,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( minus_minus_set_nat @ A4 @ B4 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ B4 ) ) ).

% Compl_Diff_eq
thf(fact_5458_case__prod__conv,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5459_case__prod__conv,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5460_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5461_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5462_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_5463_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_5464_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_5465_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_5466_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_5467_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_5468_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_5469_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_5470_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_5471_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_5472_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_5473_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_5474_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_5475_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_5476_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5477_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_5478_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_5479_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_5480_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_5481_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_5482_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_5483_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_5484_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_5485_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_5486_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_5487_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_5488_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_5489_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_5490_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_5491_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_5492_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_5493_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_5494_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_5495_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_5496_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_5497_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_5498_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_5499_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_5500_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_5501_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_5502_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_5503_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_5504_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_5505_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_5506_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_5507_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_5508_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_5509_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_5510_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5511_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5512_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( numera6690914467698888265omplex @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5513_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5514_add__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M2 ) @ ( numeral_numeral_rat @ N ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5515_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5516_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5517_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5518_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5519_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5520_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5521_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5522_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5523_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5524_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5525_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_5526_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X3 @ ( uminus6524753893492686040at_nat @ X3 ) )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_5527_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ X3 @ ( uminus_uminus_set_o @ X3 ) )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_right
thf(fact_5528_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ X3 ) )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_right
thf(fact_5529_boolean__algebra_Oconj__cancel__right,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ X3 @ ( uminus1532241313380277803et_int @ X3 ) )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_right
thf(fact_5530_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X3 ) @ X3 )
      = bot_bo2099793752762293965at_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_5531_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ X3 )
      = bot_bot_set_o ) ).

% boolean_algebra.conj_cancel_left
thf(fact_5532_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ X3 )
      = bot_bot_set_nat ) ).

% boolean_algebra.conj_cancel_left
thf(fact_5533_boolean__algebra_Oconj__cancel__left,axiom,
    ! [X3: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ X3 )
      = bot_bot_set_int ) ).

% boolean_algebra.conj_cancel_left
thf(fact_5534_inf__compl__bot__right,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ Y @ ( uminus6524753893492686040at_nat @ X3 ) ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_right
thf(fact_5535_inf__compl__bot__right,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ Y @ ( uminus_uminus_set_o @ X3 ) ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_right
thf(fact_5536_inf__compl__bot__right,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X3 ) ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_right
thf(fact_5537_inf__compl__bot__right,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X3 @ ( inf_inf_set_int @ Y @ ( uminus1532241313380277803et_int @ X3 ) ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_right
thf(fact_5538_inf__compl__bot__left2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X3 ) @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left2
thf(fact_5539_inf__compl__bot__left2,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ X3 @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left2
thf(fact_5540_inf__compl__bot__left2,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ X3 @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left2
thf(fact_5541_inf__compl__bot__left2,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ X3 @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left2
thf(fact_5542_inf__compl__bot__left1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X3 ) @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_compl_bot_left1
thf(fact_5543_inf__compl__bot__left1,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ ( inf_inf_set_o @ X3 @ Y ) )
      = bot_bot_set_o ) ).

% inf_compl_bot_left1
thf(fact_5544_inf__compl__bot__left1,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ ( inf_inf_set_nat @ X3 @ Y ) )
      = bot_bot_set_nat ) ).

% inf_compl_bot_left1
thf(fact_5545_inf__compl__bot__left1,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( inf_inf_set_int @ X3 @ Y ) )
      = bot_bot_set_int ) ).

% inf_compl_bot_left1
thf(fact_5546_subset__Compl__singleton,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A4 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5547_subset__Compl__singleton,axiom,
    ! [A4: set_complex,B: complex] :
      ( ( ord_le211207098394363844omplex @ A4 @ ( uminus8566677241136511917omplex @ ( insert_complex @ B @ bot_bot_set_complex ) ) )
      = ( ~ ( member_complex @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5548_subset__Compl__singleton,axiom,
    ! [A4: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A4 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5549_subset__Compl__singleton,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5550_subset__Compl__singleton,axiom,
    ! [A4: set_o,B: $o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ ( insert_o @ B @ bot_bot_set_o ) ) )
      = ( ~ ( member_o @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5551_subset__Compl__singleton,axiom,
    ! [A4: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5552_subset__Compl__singleton,axiom,
    ! [A4: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A4 ) ) ) ).

% subset_Compl_singleton
thf(fact_5553_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_5554_take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_5555_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_5556_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_5557_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_5558_semiring__norm_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(7)
thf(fact_5559_semiring__norm_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( bit1 @ ( plus_plus_num @ M2 @ N ) ) ) ).

% semiring_norm(9)
thf(fact_5560_semiring__norm_I15_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M2 ) @ N ) ) ) ).

% semiring_norm(15)
thf(fact_5561_semiring__norm_I14_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( bit0 @ ( times_times_num @ M2 @ ( bit1 @ N ) ) ) ) ).

% semiring_norm(14)
thf(fact_5562_semiring__norm_I72_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(72)
thf(fact_5563_semiring__norm_I81_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(81)
thf(fact_5564_semiring__norm_I70_J,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M2 ) @ one ) ).

% semiring_norm(70)
thf(fact_5565_semiring__norm_I77_J,axiom,
    ! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).

% semiring_norm(77)
thf(fact_5566_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_5567_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_5568_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_5569_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_5570_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_5571_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_5572_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_5573_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_5574_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_5575_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_5576_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_5577_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_5578_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_5579_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_5580_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_5581_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_5582_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_5583_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_5584_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_5585_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_5586_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_5587_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_5588_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_5589_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_5590_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_5591_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_5592_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_5593_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_5594_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_5595_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_5596_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_5597_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_5598_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_5599_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_5600_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_5601_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
       => ( ( ord_max_real @ ( numeral_numeral_real @ U ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
          = ( numeral_numeral_real @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5602_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
       => ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ U ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
          = ( numera6620942414471956472nteger @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5603_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
       => ( ( ord_max_rat @ ( numeral_numeral_rat @ U ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
          = ( numeral_numeral_rat @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5604_max__number__of_I2_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
       => ( ( ord_max_int @ ( numeral_numeral_int @ U ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
          = ( numeral_numeral_int @ U ) ) ) ) ).

% max_number_of(2)
thf(fact_5605_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V2 ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V2 ) )
          = ( numeral_numeral_real @ V2 ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V2 ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( numeral_numeral_real @ V2 ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5606_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V2 ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V2 ) )
          = ( numera6620942414471956472nteger @ V2 ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V2 ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( numera6620942414471956472nteger @ V2 ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5607_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V2 ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V2 ) )
          = ( numeral_numeral_rat @ V2 ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V2 ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( numeral_numeral_rat @ V2 ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5608_max__number__of_I3_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
          = ( numeral_numeral_int @ V2 ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( numeral_numeral_int @ V2 ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(3)
thf(fact_5609_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
       => ( ( ord_max_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
          = ( uminus_uminus_real @ ( numeral_numeral_real @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5610_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
       => ( ( ord_max_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) )
          = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5611_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
       => ( ( ord_max_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5612_max__number__of_I4_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
       => ( ( ord_max_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
          = ( uminus_uminus_int @ ( numeral_numeral_int @ U ) ) ) ) ) ).

% max_number_of(4)
thf(fact_5613_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5614_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5615_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V2 ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5616_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5617_semiring__norm_I168_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_5618_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5619_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5620_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5621_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5622_diff__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_5623_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5624_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5625_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5626_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5627_diff__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% diff_numeral_simps(2)
thf(fact_5628_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_5629_semiring__norm_I3_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N ) )
      = ( bit1 @ N ) ) ).

% semiring_norm(3)
thf(fact_5630_semiring__norm_I4_J,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).

% semiring_norm(4)
thf(fact_5631_semiring__norm_I5_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit0 @ M2 ) @ one )
      = ( bit1 @ M2 ) ) ).

% semiring_norm(5)
thf(fact_5632_semiring__norm_I8_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ one )
      = ( bit0 @ ( plus_plus_num @ M2 @ one ) ) ) ).

% semiring_norm(8)
thf(fact_5633_semiring__norm_I10_J,axiom,
    ! [M2: num,N: num] :
      ( ( plus_plus_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_5634_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5635_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5636_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5637_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5638_semiring__norm_I170_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_5639_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5640_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5641_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5642_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5643_semiring__norm_I171_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_5644_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5645_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5646_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5647_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5648_semiring__norm_I172_J,axiom,
    ! [V2: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_5649_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5650_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5651_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5652_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5653_mult__neg__numeral__simps_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_5654_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5655_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5656_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5657_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5658_mult__neg__numeral__simps_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_5659_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5660_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5661_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5662_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5663_mult__neg__numeral__simps_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M2 @ N ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_5664_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5665_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5666_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5667_neg__numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_eq_num @ N @ M2 ) ) ).

% neg_numeral_le_iff
thf(fact_5668_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5669_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5670_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5671_neg__numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( ord_less_num @ N @ M2 ) ) ).

% neg_numeral_less_iff
thf(fact_5672_semiring__norm_I16_J,axiom,
    ! [M2: num,N: num] :
      ( ( times_times_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M2 @ N ) @ ( bit0 @ ( times_times_num @ M2 @ N ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_5673_semiring__norm_I74_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% semiring_norm(74)
thf(fact_5674_semiring__norm_I79_J,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% semiring_norm(79)
thf(fact_5675_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5676_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5677_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5678_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M2: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) )
      = ( M2 != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5679_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5680_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5681_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5682_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5683_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5684_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5685_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5686_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5687_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5688_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5689_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5690_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5691_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5692_neg__numeral__less__neg__one__iff,axiom,
    ! [M2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M2 != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5693_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5694_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5695_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5696_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5697_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_5698_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_5699_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_5700_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_5701_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_5702_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_5703_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_5704_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_5705_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_5706_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_5707_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_5708_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_5709_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_5710_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_5711_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_5712_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_5713_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_5714_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_5715_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_5716_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_5717_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_5718_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_5719_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_5720_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_5721_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_5722_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_5723_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_5724_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_5725_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_5726_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_5727_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_5728_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_5729_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_5730_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_5731_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_5732_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_5733_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_5734_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_5735_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_5736_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_5737_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_5738_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_5739_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_5740_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_5741_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_5742_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_5743_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_5744_even__take__bit__eq,axiom,
    ! [N: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_5745_even__take__bit__eq,axiom,
    ! [N: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_5746_even__take__bit__eq,axiom,
    ! [N: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( ( N = zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_5747_Suc__div__eq__add3__div__numeral,axiom,
    ! [M2: nat,V2: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V2 ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V2 ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_5748_div__Suc__eq__div__add3,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( divide_divide_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_5749_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_5750_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_5751_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_5752_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_5753_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_5754_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5755_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5756_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5757_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5758_diff__numeral__special_I4_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M2 @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_5759_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M2: nat,V2: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral_nat @ V2 ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ ( numeral_numeral_nat @ V2 ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_5760_mod__Suc__eq__mod__add3,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
      = ( modulo_modulo_nat @ M2 @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_5761_signed__take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_Suc_minus_bit0
thf(fact_5762_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_5763_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_5764_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5765_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_5766_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_5767_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_5768_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_5769_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_5770_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_5771_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_5772_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_5773_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_5774_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_5775_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_5776_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_5777_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_5778_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_5779_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_5780_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_5781_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_5782_take__bit__Suc__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_5783_take__bit__Suc__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_5784_take__bit__Suc__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_5785_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_5786_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_5787_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_5788_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_5789_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_5790_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_5791_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_5792_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_5793_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se1745604003318907178nteger @ M2 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N @ M2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_5794_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2925701944663578781it_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_5795_take__bit__of__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( bit_se2923211474154528505it_int @ M2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N @ M2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_of_exp
thf(fact_5796_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ N @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_5797_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_5798_take__bit__of__2,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_5799_zmod__numeral__Bit1,axiom,
    ! [V2: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_5800_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_5801_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_5802_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_5803_signed__take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_Suc_minus_bit1
thf(fact_5804_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5805_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5806_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_5807_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5808_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5809_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5810_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5811_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5812_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5813_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5814_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5815_prod_Ocase__distrib,axiom,
    ! [H: $o > nat,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5816_prod_Ocase__distrib,axiom,
    ! [H: nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5817_prod_Ocase__distrib,axiom,
    ! [H: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5818_prod_Ocase__distrib,axiom,
    ! [H: $o > product_prod_nat_nat > $o,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5819_prod_Ocase__distrib,axiom,
    ! [H: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5820_prod_Ocase__distrib,axiom,
    ! [H: ( product_prod_nat_nat > $o ) > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc6081775807080527818_nat_o
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5821_prod_Ocase__distrib,axiom,
    ! [H: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5822_prod_Ocase__distrib,axiom,
    ! [H: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X23: int] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5823_prod_Ocase__distrib,axiom,
    ! [H: $o > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > $o,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc6081775807080527818_nat_o @ F @ Prod ) )
      = ( produc27273713700761075at_nat
        @ ^ [X15: nat,X23: nat] : ( H @ ( F @ X15 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_5824_power__minus__Bit1,axiom,
    ! [X3: int,K2: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5825_power__minus__Bit1,axiom,
    ! [X3: real,K2: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5826_power__minus__Bit1,axiom,
    ! [X3: complex,K2: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5827_power__minus__Bit1,axiom,
    ! [X3: code_integer,K2: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5828_power__minus__Bit1,axiom,
    ! [X3: rat,K2: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_5829_take__bit__add,axiom,
    ! [N: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N @ A ) @ ( bit_se2925701944663578781it_nat @ N @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_5830_take__bit__add,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ A ) @ ( bit_se2923211474154528505it_int @ N @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_5831_take__bit__tightened,axiom,
    ! [N: nat,A: nat,B: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = ( bit_se2925701944663578781it_nat @ N @ B ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( bit_se2925701944663578781it_nat @ M2 @ A )
          = ( bit_se2925701944663578781it_nat @ M2 @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_5832_take__bit__tightened,axiom,
    ! [N: nat,A: int,B: int,M2: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = ( bit_se2923211474154528505it_int @ N @ B ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( bit_se2923211474154528505it_int @ M2 @ A )
          = ( bit_se2923211474154528505it_int @ M2 @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_5833_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_5834_take__bit__nat__less__eq__self,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 ) ).

% take_bit_nat_less_eq_self
thf(fact_5835_compl__le__swap2,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X3 )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5836_compl__le__swap1,axiom,
    ! [Y: set_nat,X3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X3 ) )
     => ( ord_less_eq_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5837_compl__mono,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X3 @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X3 ) ) ) ).

% compl_mono
thf(fact_5838_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5839_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5840_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5841_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5842_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5843_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5844_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5845_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5846_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_5847_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5848_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5849_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_5850_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_5851_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_5852_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5853_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5854_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5855_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5856_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5857_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5858_verit__eq__simplify_I14_J,axiom,
    ! [X2: num,X32: num] :
      ( ( bit0 @ X2 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_5859_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_5860_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_int @ M2 )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5861_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_real @ M2 )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5862_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6690914467698888265omplex @ M2 )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5863_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6620942414471956472nteger @ M2 )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5864_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_rat @ M2 )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5865_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) )
     != ( numeral_numeral_int @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5866_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) )
     != ( numeral_numeral_real @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5867_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5868_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5869_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) )
     != ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_5870_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5871_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5872_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5873_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5874_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5875_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5876_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5877_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5878_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5879_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5880_group__cancel_Oneg1,axiom,
    ! [A4: int,K2: int,A: int] :
      ( ( A4
        = ( plus_plus_int @ K2 @ A ) )
     => ( ( uminus_uminus_int @ A4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( uminus_uminus_int @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5881_group__cancel_Oneg1,axiom,
    ! [A4: real,K2: real,A: real] :
      ( ( A4
        = ( plus_plus_real @ K2 @ A ) )
     => ( ( uminus_uminus_real @ A4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( uminus_uminus_real @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5882_group__cancel_Oneg1,axiom,
    ! [A4: complex,K2: complex,A: complex] :
      ( ( A4
        = ( plus_plus_complex @ K2 @ A ) )
     => ( ( uminus1482373934393186551omplex @ A4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5883_group__cancel_Oneg1,axiom,
    ! [A4: code_integer,K2: code_integer,A: code_integer] :
      ( ( A4
        = ( plus_p5714425477246183910nteger @ K2 @ A ) )
     => ( ( uminus1351360451143612070nteger @ A4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5884_group__cancel_Oneg1,axiom,
    ! [A4: rat,K2: rat,A: rat] :
      ( ( A4
        = ( plus_plus_rat @ K2 @ A ) )
     => ( ( uminus_uminus_rat @ A4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5885_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5886_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5887_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5888_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5889_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5890_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > $o,X1: nat,X2: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_5891_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_5892_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_5893_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_5894_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_5895_case__prodE2,axiom,
    ! [Q: $o > $o,P: nat > nat > $o,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc6081775807080527818_nat_o @ P @ Z2 ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5896_case__prodE2,axiom,
    ! [Q: nat > $o,P: nat > nat > nat,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z2 ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5897_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5898_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5899_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
     => ~ ! [X5: int,Y4: int] :
            ( ( Z2
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( Q @ ( P @ X5 @ Y4 ) ) ) ) ).

% case_prodE2
thf(fact_5900_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5901_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5902_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5903_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y3: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5904_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X4: int,Y3: int] : ( F @ ( product_Pair_int_int @ X4 @ Y3 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_5905_cond__case__prod__eta,axiom,
    ! [F: nat > nat > $o,G: product_prod_nat_nat > $o] :
      ( ! [X5: nat,Y4: nat] :
          ( ( F @ X5 @ Y4 )
          = ( G @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) )
     => ( ( produc6081775807080527818_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_5906_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_5907_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_5908_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_5909_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_5910_Collect__imp__eq,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( collect_complex
        @ ^ [X4: complex] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_complex @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) @ ( collect_complex @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5911_Collect__imp__eq,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) @ ( collec3392354462482085612at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5912_Collect__imp__eq,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_set_nat @ ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5913_Collect__imp__eq,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_list_nat @ ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5914_Collect__imp__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) @ ( collect_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5915_Collect__imp__eq,axiom,
    ! [P: produc3843707927480180839at_nat > $o,Q: produc3843707927480180839at_nat > $o] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X4: produc3843707927480180839at_nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ ( collec6321179662152712658at_nat @ P ) ) @ ( collec6321179662152712658at_nat @ Q ) ) ) ).

% Collect_imp_eq
thf(fact_5916_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_5917_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_5918_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_5919_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_5920_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_5921_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_5922_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_5923_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_5924_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_5925_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_5926_take__bit__tightened__less__eq__int,axiom,
    ! [M2: nat,N: nat,K2: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M2 @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_5927_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ A )
        = ( bit_ri631733984087533419it_int @ N @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_5928_signed__take__bit__take__bit,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N @ M2 ) @ ( bit_se2923211474154528505it_int @ N ) @ ( bit_ri631733984087533419it_int @ M2 ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_5929_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5930_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5931_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5932_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5933_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5934_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5935_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5936_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_5937_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5938_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5939_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5940_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5941_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5942_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5943_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( numeral_numeral_real @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5944_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5945_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( numeral_numeral_rat @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_5946_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5947_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5948_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5949_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5950_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X22: num] :
            ( Y
           != ( bit0 @ X22 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_5951_xor__num_Ocases,axiom,
    ! [X3: product_prod_num_num] :
      ( ( X3
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N2: num] :
            ( X3
           != ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) )
       => ( ! [N2: num] :
              ( X3
             != ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) )
         => ( ! [M: num] :
                ( X3
               != ( product_Pair_num_num @ ( bit0 @ M ) @ one ) )
           => ( ! [M: num,N2: num] :
                  ( X3
                 != ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) )
             => ( ! [M: num,N2: num] :
                    ( X3
                   != ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) )
               => ( ! [M: num] :
                      ( X3
                     != ( product_Pair_num_num @ ( bit1 @ M ) @ one ) )
                 => ( ! [M: num,N2: num] :
                        ( X3
                       != ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) )
                   => ~ ! [M: num,N2: num] :
                          ( X3
                         != ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5961_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5962_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5963_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5964_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5965_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_5966_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_5967_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_5968_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_5969_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_5970_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5971_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5972_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5973_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5974_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5975_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5976_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5977_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5978_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5979_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5980_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5981_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5982_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5983_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5984_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5985_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_5986_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_5987_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_5988_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_5989_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_5990_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5991_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5992_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5993_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5994_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5995_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5996_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5997_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5998_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_5999_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_6000_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A6: int,B7: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6001_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A6: real,B7: real] : ( plus_plus_real @ A6 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6002_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A6: complex,B7: complex] : ( plus_plus_complex @ A6 @ ( uminus1482373934393186551omplex @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6003_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A6: code_integer,B7: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6004_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A6: rat,B7: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_6005_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A6: int,B7: int] : ( plus_plus_int @ A6 @ ( uminus_uminus_int @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6006_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A6: real,B7: real] : ( plus_plus_real @ A6 @ ( uminus_uminus_real @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6007_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A6: complex,B7: complex] : ( plus_plus_complex @ A6 @ ( uminus1482373934393186551omplex @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6008_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A6: code_integer,B7: code_integer] : ( plus_p5714425477246183910nteger @ A6 @ ( uminus1351360451143612070nteger @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6009_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A6: rat,B7: rat] : ( plus_plus_rat @ A6 @ ( uminus_uminus_rat @ B7 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_6010_group__cancel_Osub2,axiom,
    ! [B4: int,K2: int,B: int,A: int] :
      ( ( B4
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( minus_minus_int @ A @ B4 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6011_group__cancel_Osub2,axiom,
    ! [B4: real,K2: real,B: real,A: real] :
      ( ( B4
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( minus_minus_real @ A @ B4 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6012_group__cancel_Osub2,axiom,
    ! [B4: complex,K2: complex,B: complex,A: complex] :
      ( ( B4
        = ( plus_plus_complex @ K2 @ B ) )
     => ( ( minus_minus_complex @ A @ B4 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6013_group__cancel_Osub2,axiom,
    ! [B4: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B4
        = ( plus_p5714425477246183910nteger @ K2 @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B4 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6014_group__cancel_Osub2,axiom,
    ! [B4: rat,K2: rat,B: rat,A: rat] :
      ( ( B4
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( minus_minus_rat @ A @ B4 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_6015_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se4205575877204974255it_nat @ M2 @ A ) )
          = ( bit_se4205575877204974255it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6016_take__bit__unset__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se4203085406695923979it_int @ M2 @ A ) )
          = ( bit_se4203085406695923979it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_6017_take__bit__set__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se7882103937844011126it_nat @ M2 @ A ) )
          = ( bit_se7882103937844011126it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6018_take__bit__set__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se7879613467334960850it_int @ M2 @ A ) )
          = ( bit_se7879613467334960850it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_6019_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M2: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
          = ( bit_se2925701944663578781it_nat @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2925701944663578781it_nat @ N @ ( bit_se2161824704523386999it_nat @ M2 @ A ) )
          = ( bit_se2161824704523386999it_nat @ M2 @ ( bit_se2925701944663578781it_nat @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6020_take__bit__flip__bit__eq,axiom,
    ! [N: nat,M2: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
          = ( bit_se2923211474154528505it_int @ N @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N @ M2 )
       => ( ( bit_se2923211474154528505it_int @ N @ ( bit_se2159334234014336723it_int @ M2 @ A ) )
          = ( bit_se2159334234014336723it_int @ M2 @ ( bit_se2923211474154528505it_int @ N @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_6021_inf__cancel__left2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X3 ) @ A ) @ ( inf_in2572325071724192079at_nat @ X3 @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left2
thf(fact_6022_inf__cancel__left2,axiom,
    ! [X3: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ A ) @ ( inf_inf_set_o @ X3 @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left2
thf(fact_6023_inf__cancel__left2,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ A ) @ ( inf_inf_set_nat @ X3 @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left2
thf(fact_6024_inf__cancel__left2,axiom,
    ! [X3: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ A ) @ ( inf_inf_set_int @ X3 @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left2
thf(fact_6025_inf__cancel__left1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( inf_in2572325071724192079at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ A ) @ ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ X3 ) @ B ) )
      = bot_bo2099793752762293965at_nat ) ).

% inf_cancel_left1
thf(fact_6026_inf__cancel__left1,axiom,
    ! [X3: set_o,A: set_o,B: set_o] :
      ( ( inf_inf_set_o @ ( inf_inf_set_o @ X3 @ A ) @ ( inf_inf_set_o @ ( uminus_uminus_set_o @ X3 ) @ B ) )
      = bot_bot_set_o ) ).

% inf_cancel_left1
thf(fact_6027_inf__cancel__left1,axiom,
    ! [X3: set_nat,A: set_nat,B: set_nat] :
      ( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X3 @ A ) @ ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ X3 ) @ B ) )
      = bot_bot_set_nat ) ).

% inf_cancel_left1
thf(fact_6028_inf__cancel__left1,axiom,
    ! [X3: set_int,A: set_int,B: set_int] :
      ( ( inf_inf_set_int @ ( inf_inf_set_int @ X3 @ A ) @ ( inf_inf_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ B ) )
      = bot_bot_set_int ) ).

% inf_cancel_left1
thf(fact_6029_subset__Compl__self__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ A4 ) )
      = ( A4 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_6030_subset__Compl__self__eq,axiom,
    ! [A4: set_o] :
      ( ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ A4 ) )
      = ( A4 = bot_bot_set_o ) ) ).

% subset_Compl_self_eq
thf(fact_6031_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_6032_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_6033_Compl__Int,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) )
      = ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( uminus6524753893492686040at_nat @ B4 ) ) ) ).

% Compl_Int
thf(fact_6034_Compl__Int,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( inf_inf_set_nat @ A4 @ B4 ) )
      = ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).

% Compl_Int
thf(fact_6035_Compl__Int,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) )
      = ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( uminus935396558254630718at_nat @ B4 ) ) ) ).

% Compl_Int
thf(fact_6036_Compl__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
      = ( inf_in2572325071724192079at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( uminus6524753893492686040at_nat @ B4 ) ) ) ).

% Compl_Un
thf(fact_6037_Compl__Un,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( inf_inf_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).

% Compl_Un
thf(fact_6038_Compl__Un,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
      = ( inf_in7913087082777306421at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( uminus935396558254630718at_nat @ B4 ) ) ) ).

% Compl_Un
thf(fact_6039_Diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] : ( inf_in2572325071724192079at_nat @ A5 @ ( uminus6524753893492686040at_nat @ B5 ) ) ) ) ).

% Diff_eq
thf(fact_6040_Diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ A5 @ ( uminus5710092332889474511et_nat @ B5 ) ) ) ) ).

% Diff_eq
thf(fact_6041_take__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_minus_bit0
thf(fact_6042_take__bit__signed__take__bit,axiom,
    ! [M2: nat,N: nat,A: int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( bit_se2923211474154528505it_int @ M2 @ ( bit_ri631733984087533419it_int @ N @ A ) )
        = ( bit_se2923211474154528505it_int @ M2 @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_6043_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_6044_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_6045_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_6046_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_6047_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_6048_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_6049_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_6050_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_6051_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_6052_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_6053_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_6054_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_6055_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_6056_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_6057_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_6058_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_6059_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_6060_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_6061_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_6062_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_6063_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_6064_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_6065_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_6066_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_6067_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_6068_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_6069_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_6070_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_6071_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_6072_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6073_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6074_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6075_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_6076_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_6077_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_6078_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_6079_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_6080_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_6081_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_6082_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_6083_neg__numeral__le__neg__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_6084_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6085_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6086_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6087_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_6088_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_6089_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_6090_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_6091_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_6092_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_6093_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_6094_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_6095_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_6096_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6097_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6098_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6099_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_6100_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_6101_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_6102_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_6103_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_6104_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6105_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6106_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6107_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_6108_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6109_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6110_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6111_not__neg__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_6112_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6113_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6114_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6115_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6116_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_6117_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6118_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6119_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6120_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6121_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_6122_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_6123_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_6124_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_6125_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_6126_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_6127_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_6128_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_6129_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_6130_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_6131_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_6132_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_6133_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6134_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6135_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_6136_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6137_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6138_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_6139_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6140_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6141_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_6142_inf__shunt,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X3 @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X3 @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_6143_inf__shunt,axiom,
    ! [X3: set_o,Y: set_o] :
      ( ( ( inf_inf_set_o @ X3 @ Y )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ X3 @ ( uminus_uminus_set_o @ Y ) ) ) ).

% inf_shunt
thf(fact_6144_inf__shunt,axiom,
    ! [X3: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X3 @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% inf_shunt
thf(fact_6145_inf__shunt,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X3 @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_6146_power__minus__Bit0,axiom,
    ! [X3: int,K2: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_6147_power__minus__Bit0,axiom,
    ! [X3: real,K2: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_6148_power__minus__Bit0,axiom,
    ! [X3: complex,K2: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_6149_power__minus__Bit0,axiom,
    ! [X3: code_integer,K2: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_6150_power__minus__Bit0,axiom,
    ! [X3: rat,K2: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ) ).

% power_minus_Bit0
thf(fact_6151_sup__neg__inf,axiom,
    ! [P2: set_Pr1261947904930325089at_nat,Q3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P2 @ ( sup_su6327502436637775413at_nat @ Q3 @ R2 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P2 @ ( uminus6524753893492686040at_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6152_sup__neg__inf,axiom,
    ! [P2: set_Pr4329608150637261639at_nat,Q3: set_Pr4329608150637261639at_nat,R2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ P2 @ ( sup_su5525570899277871387at_nat @ Q3 @ R2 ) )
      = ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ P2 @ ( uminus935396558254630718at_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6153_sup__neg__inf,axiom,
    ! [P2: set_nat,Q3: set_nat,R2: set_nat] :
      ( ( ord_less_eq_set_nat @ P2 @ ( sup_sup_set_nat @ Q3 @ R2 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P2 @ ( uminus5710092332889474511et_nat @ Q3 ) ) @ R2 ) ) ).

% sup_neg_inf
thf(fact_6154_shunt2,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z2 )
      = ( ord_le3146513528884898305at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ Y @ Z2 ) ) ) ).

% shunt2
thf(fact_6155_shunt2,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ ( uminus935396558254630718at_nat @ Y ) ) @ Z2 )
      = ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ Y @ Z2 ) ) ) ).

% shunt2
thf(fact_6156_shunt2,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z2 )
      = ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).

% shunt2
thf(fact_6157_shunt1,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X3 @ Y ) @ Z2 )
      = ( ord_le3146513528884898305at_nat @ X3 @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z2 ) ) ) ).

% shunt1
thf(fact_6158_shunt1,axiom,
    ! [X3: set_Pr4329608150637261639at_nat,Y: set_Pr4329608150637261639at_nat,Z2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( inf_in7913087082777306421at_nat @ X3 @ Y ) @ Z2 )
      = ( ord_le1268244103169919719at_nat @ X3 @ ( sup_su5525570899277871387at_nat @ ( uminus935396558254630718at_nat @ Y ) @ Z2 ) ) ) ).

% shunt1
thf(fact_6159_shunt1,axiom,
    ! [X3: set_nat,Y: set_nat,Z2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X3 @ Y ) @ Z2 )
      = ( ord_less_eq_set_nat @ X3 @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z2 ) ) ) ).

% shunt1
thf(fact_6160_take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_Suc_bit1
thf(fact_6161_take__bit__Suc__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_bit1
thf(fact_6162_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ A4 @ B4 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A4 @ ( uminus6524753893492686040at_nat @ B4 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6163_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_o,B4: set_o] :
      ( ( ( inf_inf_set_o @ A4 @ B4 )
        = bot_bot_set_o )
      = ( ord_less_eq_set_o @ A4 @ ( uminus_uminus_set_o @ B4 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6164_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_int,B4: set_int] :
      ( ( ( inf_inf_set_int @ A4 @ B4 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A4 @ ( uminus1532241313380277803et_int @ B4 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6165_disjoint__eq__subset__Compl,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ( ( inf_inf_set_nat @ A4 @ B4 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ B4 ) ) ) ).

% disjoint_eq_subset_Compl
thf(fact_6166_Compl__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X3 @ A4 ) )
      = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A4 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% Compl_insert
thf(fact_6167_Compl__insert,axiom,
    ! [X3: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X3 @ A4 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A4 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_6168_Compl__insert,axiom,
    ! [X3: $o,A4: set_o] :
      ( ( uminus_uminus_set_o @ ( insert_o @ X3 @ A4 ) )
      = ( minus_minus_set_o @ ( uminus_uminus_set_o @ A4 ) @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ).

% Compl_insert
thf(fact_6169_Compl__insert,axiom,
    ! [X3: int,A4: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A4 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A4 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_6170_Compl__insert,axiom,
    ! [X3: nat,A4: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A4 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A4 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_6171_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_6172_take__bit__Suc__minus__1__eq,axiom,
    ! [N: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_6173_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K2 ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_6174_take__bit__numeral__minus__1__eq,axiom,
    ! [K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K2 ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_6175_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_6176_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_6177_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_6178_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_6179_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_6180_power__numeral__odd,axiom,
    ! [Z2: complex,W: num] :
      ( ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z2 @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6181_power__numeral__odd,axiom,
    ! [Z2: real,W: num] :
      ( ( power_power_real @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z2 @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6182_power__numeral__odd,axiom,
    ! [Z2: rat,W: num] :
      ( ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z2 @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_rat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6183_power__numeral__odd,axiom,
    ! [Z2: nat,W: num] :
      ( ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z2 @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6184_power__numeral__odd,axiom,
    ! [Z2: int,W: num] :
      ( ( power_power_int @ Z2 @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z2 @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z2 @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_6185_take__bit__minus__small__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ K2 ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_6186_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_6187_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_6188_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6189_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6190_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_6191_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6192_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6193_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_6194_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6195_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6196_odd__numeral,axiom,
    ! [N: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) ).

% odd_numeral
thf(fact_6197_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_6198_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_6199_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_6200_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6201_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6202_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_6203_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X3: real,Y: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6204_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6205_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_6206_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_6207_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_6208_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_6209_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_6210_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_6211_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_6212_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_6213_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_6214_power2__eq__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y )
        | ( X3
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6215_power2__eq__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y )
        | ( X3
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6216_power2__eq__iff,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y )
        | ( X3
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6217_power2__eq__iff,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y )
        | ( X3
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6218_power2__eq__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X3 = Y )
        | ( X3
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_6219_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_6220_take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6221_take__bit__Suc__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_bit0
thf(fact_6222_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N3: nat,A6: code_integer] : ( modulo364778990260209775nteger @ A6 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6223_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,A6: nat] : ( modulo_modulo_nat @ A6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6224_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,A6: int] : ( modulo_modulo_int @ A6 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_eq_mod
thf(fact_6225_take__bit__nat__eq__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 )
      = ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_6226_take__bit__nat__less__exp,axiom,
    ! [N: nat,M2: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_nat_less_exp
thf(fact_6227_take__bit__nat__eq__self,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( bit_se2925701944663578781it_nat @ N @ M2 )
        = M2 ) ) ).

% take_bit_nat_eq_self
thf(fact_6228_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_6229_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N3: nat,M5: nat] : ( modulo_modulo_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_nat_def
thf(fact_6230_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_6231_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_6232_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_6233_take__bit__int__less__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ).

% take_bit_int_less_exp
thf(fact_6234_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6235_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_6236_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6237_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_6238_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6239_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_6240_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6241_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_6242_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6243_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_6244_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6245_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_6246_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_6247_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_6248_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_6249_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_6250_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N3: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% take_bit_int_def
thf(fact_6251_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6252_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6253_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_6254_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6255_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6256_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_6257_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_6258_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_6259_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_6260_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_6261_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_6262_Suc__div__eq__add3__div,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).

% Suc_div_eq_add3_div
thf(fact_6263_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_6264_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_6265_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_6266_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_6267_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_6268_Suc__mod__eq__add3__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M2 ) @ N ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_6269_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6270_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6271_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6272_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6273_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N @ K2 ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_6274_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_6275_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_6276_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6277_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N @ A )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6278_take__bit__eq__0__iff,axiom,
    ! [N: nat,A: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ A )
        = zero_zero_int )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_6279_take__bit__nat__less__self__iff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N @ M2 ) @ M2 )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 ) ) ).

% take_bit_nat_less_self_iff
thf(fact_6280_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_6281_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_6282_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_6283_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_6284_take__bit__int__greater__eq__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ K2 @ ( bit_se2923211474154528505it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_6285_take__bit__int__less__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 ) ) ).

% take_bit_int_less_self_iff
thf(fact_6286_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_6287_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_6288_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_6289_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_6290_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_6291_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_6292_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_6293_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_6294_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_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_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_6304_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_6305_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N: nat,K2: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_6306_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ K2 )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_6307_signed__take__bit__int__greater__self__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ ( bit_ri631733984087533419it_int @ N @ K2 ) )
      = ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_6308_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_6309_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M2 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_6310_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M2 @ N ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_6311_take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_6312_take__bit__int__eq__self,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_se2923211474154528505it_int @ N @ K2 )
          = K2 ) ) ) ).

% take_bit_int_eq_self
thf(fact_6313_take__bit__incr__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N @ ( plus_plus_int @ K2 @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ) ).

% take_bit_incr_eq
thf(fact_6314_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_6315_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_6316_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_6317_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_6318_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_6319_take__bit__Suc,axiom,
    ! [N: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6320_take__bit__Suc,axiom,
    ! [N: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N ) @ A )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6321_take__bit__Suc,axiom,
    ! [N: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_6322_int__bit__induct,axiom,
    ! [P: int > $o,K2: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K: int] :
              ( ( P @ K )
             => ( ( K != zero_zero_int )
               => ( P @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K: int] :
                ( ( P @ K )
               => ( ( K
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K2 ) ) ) ) ) ).

% int_bit_induct
thf(fact_6323_signed__take__bit__int__eq__self__iff,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_ri631733984087533419it_int @ N @ K2 )
        = K2 )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
        & ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_6324_signed__take__bit__int__eq__self,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ K2 )
     => ( ( ord_less_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
       => ( ( bit_ri631733984087533419it_int @ N @ K2 )
          = K2 ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_6325_take__bit__int__less__eq,axiom,
    ! [N: nat,K2: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ K2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( minus_minus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% take_bit_int_less_eq
thf(fact_6326_take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_int_greater_eq
thf(fact_6327_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_6328_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6329_stable__imp__take__bit__eq,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = zero_zero_nat ) )
        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2925701944663578781it_nat @ N @ A )
            = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6330_stable__imp__take__bit__eq,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = zero_zero_int ) )
        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se2923211474154528505it_int @ N @ A )
            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_6331_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_6332_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_6333_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6334_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6335_exp__div__exp__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M2 )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N @ M2 ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% exp_div_exp_eq
thf(fact_6336_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_6337_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_6338_signed__take__bit__int__greater__eq,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_int @ K2 @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ K2 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) ) @ ( bit_ri631733984087533419it_int @ N @ K2 ) ) ) ).

% signed_take_bit_int_greater_eq
thf(fact_6339_mod__exhaust__less__4,axiom,
    ! [M2: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_6340_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_6341_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6342_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6343_divmod__algorithm__code_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M2 @ N ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6344_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_6345_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_6346_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6347_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_6348_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_6349_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6350_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6351_divmod__algorithm__code_I7_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_6352_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6353_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N ) @ ( unique5052692396658037445od_int @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6354_divmod__algorithm__code_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M2 ) ) ) ) )
      & ( ~ ( ord_less_num @ M2 @ N )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N ) @ ( unique3479559517661332726nteger @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_6355_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_6356_ComplI,axiom,
    ! [C: complex,A4: set_complex] :
      ( ~ ( member_complex @ C @ A4 )
     => ( member_complex @ C @ ( uminus8566677241136511917omplex @ A4 ) ) ) ).

% ComplI
thf(fact_6357_ComplI,axiom,
    ! [C: real,A4: set_real] :
      ( ~ ( member_real @ C @ A4 )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A4 ) ) ) ).

% ComplI
thf(fact_6358_ComplI,axiom,
    ! [C: $o,A4: set_o] :
      ( ~ ( member_o @ C @ A4 )
     => ( member_o @ C @ ( uminus_uminus_set_o @ A4 ) ) ) ).

% ComplI
thf(fact_6359_ComplI,axiom,
    ! [C: nat,A4: set_nat] :
      ( ~ ( member_nat @ C @ A4 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A4 ) ) ) ).

% ComplI
thf(fact_6360_ComplI,axiom,
    ! [C: int,A4: set_int] :
      ( ~ ( member_int @ C @ A4 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A4 ) ) ) ).

% ComplI
thf(fact_6361_Compl__iff,axiom,
    ! [C: complex,A4: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A4 ) )
      = ( ~ ( member_complex @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_6362_Compl__iff,axiom,
    ! [C: real,A4: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A4 ) )
      = ( ~ ( member_real @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_6363_Compl__iff,axiom,
    ! [C: $o,A4: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A4 ) )
      = ( ~ ( member_o @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_6364_Compl__iff,axiom,
    ! [C: nat,A4: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A4 ) )
      = ( ~ ( member_nat @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_6365_Compl__iff,axiom,
    ! [C: int,A4: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A4 ) )
      = ( ~ ( member_int @ C @ A4 ) ) ) ).

% Compl_iff
thf(fact_6366_case__prodI2,axiom,
    ! [P2: produc3843707927480180839at_nat,C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
          ( ( P2
            = ( produc2922128104949294807at_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc410239310623530412_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_6367_case__prodI2,axiom,
    ! [P2: product_prod_num_num,C: num > num > $o] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc5703948589228662326_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_6368_case__prodI2,axiom,
    ! [P2: product_prod_nat_num,C: nat > num > $o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4927758841916487424_num_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_6369_case__prodI2,axiom,
    ! [P2: product_prod_int_int,C: int > int > $o] :
      ( ! [A3: int,B3: int] :
          ( ( P2
            = ( product_Pair_int_int @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc4947309494688390418_int_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_6370_case__prodI2,axiom,
    ! [P2: product_prod_nat_nat,C: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C @ P2 ) ) ).

% case_prodI2
thf(fact_6371_case__prodI,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( F @ A @ B )
     => ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_6372_case__prodI,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( F @ A @ B )
     => ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_6373_case__prodI,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( F @ A @ B )
     => ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) ) ) ).

% case_prodI
thf(fact_6374_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_6375_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_6376_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: complex,C: num > num > set_complex] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_complex @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z2 @ ( produc2866383454006189126omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6377_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: real,C: num > num > set_real] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z2 @ ( produc8296048397933160132t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6378_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: $o,C: num > num > set_o] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_o @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_o @ Z2 @ ( produc1904233663560707350_set_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6379_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: nat,C: num > num > set_nat] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6380_mem__case__prodI2,axiom,
    ! [P2: product_prod_num_num,Z2: int,C: num > num > set_int] :
      ( ! [A3: num,B3: num] :
          ( ( P2
            = ( product_Pair_num_num @ A3 @ B3 ) )
         => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z2 @ ( produc6406642877701697732et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6381_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: complex,C: nat > num > set_complex] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_complex @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_complex @ Z2 @ ( produc6231982587499038204omplex @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6382_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: real,C: nat > num > set_real] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_real @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_real @ Z2 @ ( produc1435849484188172666t_real @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6383_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: $o,C: nat > num > set_o] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_o @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6384_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: nat,C: nat > num > set_nat] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6385_mem__case__prodI2,axiom,
    ! [P2: product_prod_nat_num,Z2: int,C: nat > num > set_int] :
      ( ! [A3: nat,B3: num] :
          ( ( P2
            = ( product_Pair_nat_num @ A3 @ B3 ) )
         => ( member_int @ Z2 @ ( C @ A3 @ B3 ) ) )
     => ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_6386_mem__case__prodI,axiom,
    ! [Z2: complex,C: num > num > set_complex,A: num,B: num] :
      ( ( member_complex @ Z2 @ ( C @ A @ B ) )
     => ( member_complex @ Z2 @ ( produc2866383454006189126omplex @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6387_mem__case__prodI,axiom,
    ! [Z2: real,C: num > num > set_real,A: num,B: num] :
      ( ( member_real @ Z2 @ ( C @ A @ B ) )
     => ( member_real @ Z2 @ ( produc8296048397933160132t_real @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6388_mem__case__prodI,axiom,
    ! [Z2: $o,C: num > num > set_o,A: num,B: num] :
      ( ( member_o @ Z2 @ ( C @ A @ B ) )
     => ( member_o @ Z2 @ ( produc1904233663560707350_set_o @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6389_mem__case__prodI,axiom,
    ! [Z2: nat,C: num > num > set_nat,A: num,B: num] :
      ( ( member_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6390_mem__case__prodI,axiom,
    ! [Z2: int,C: num > num > set_int,A: num,B: num] :
      ( ( member_int @ Z2 @ ( C @ A @ B ) )
     => ( member_int @ Z2 @ ( produc6406642877701697732et_int @ C @ ( product_Pair_num_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6391_mem__case__prodI,axiom,
    ! [Z2: complex,C: nat > num > set_complex,A: nat,B: num] :
      ( ( member_complex @ Z2 @ ( C @ A @ B ) )
     => ( member_complex @ Z2 @ ( produc6231982587499038204omplex @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6392_mem__case__prodI,axiom,
    ! [Z2: real,C: nat > num > set_real,A: nat,B: num] :
      ( ( member_real @ Z2 @ ( C @ A @ B ) )
     => ( member_real @ Z2 @ ( produc1435849484188172666t_real @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6393_mem__case__prodI,axiom,
    ! [Z2: $o,C: nat > num > set_o,A: nat,B: num] :
      ( ( member_o @ Z2 @ ( C @ A @ B ) )
     => ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6394_mem__case__prodI,axiom,
    ! [Z2: nat,C: nat > num > set_nat,A: nat,B: num] :
      ( ( member_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6395_mem__case__prodI,axiom,
    ! [Z2: int,C: nat > num > set_int,A: nat,B: num] :
      ( ( member_int @ Z2 @ ( C @ A @ B ) )
     => ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ ( product_Pair_nat_num @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6396_case__prodI2_H,axiom,
    ! [P2: 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 )
            = P2 )
         => ( C @ A3 @ B3 @ X3 ) )
     => ( produc8739625826339149834_nat_o @ C @ P2 @ X3 ) ) ).

% case_prodI2'
thf(fact_6397_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6398_eq__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K2 )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K2 )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_6399_Suc__eq__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K2 ) )
      = ( N
        = ( pred_numeral @ K2 ) ) ) ).

% Suc_eq_numeral
thf(fact_6400_less__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% less_numeral_Suc
thf(fact_6401_less__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_less_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% less_Suc_numeral
thf(fact_6402_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit1 @ K2 ) )
      = ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ).

% pred_numeral_simps(3)
thf(fact_6403_le__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( ord_less_eq_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% le_Suc_numeral
thf(fact_6404_le__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( ord_less_eq_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% le_numeral_Suc
thf(fact_6405_diff__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ ( pred_numeral @ K2 ) @ N ) ) ).

% diff_numeral_Suc
thf(fact_6406_diff__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( minus_minus_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( minus_minus_nat @ N @ ( pred_numeral @ K2 ) ) ) ).

% diff_Suc_numeral
thf(fact_6407_max__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_max_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).

% max_numeral_Suc
thf(fact_6408_max__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_max_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( suc @ ( ord_max_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).

% max_Suc_numeral
thf(fact_6409_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6410_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6411_dvd__numeral__simp,axiom,
    ! [M2: num,N: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N @ M2 ) ) ) ).

% dvd_numeral_simp
thf(fact_6412_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ M2 @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M2 ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_6413_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique5055182867167087721od_nat @ M2 @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M2 ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_6414_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique3479559517661332726nteger @ M2 @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M2 ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_6415_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_6416_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_6417_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_6418_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_6419_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_6420_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_6421_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_6422_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q4: int,R5: int] : ( product_Pair_int_int @ Q4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6423_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q4: nat,R5: nat] : ( product_Pair_nat_nat @ Q4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M2 @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6424_divmod__algorithm__code_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q4: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M2 @ N ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6425_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_6426_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_6427_ComplD,axiom,
    ! [C: complex,A4: set_complex] :
      ( ( member_complex @ C @ ( uminus8566677241136511917omplex @ A4 ) )
     => ~ ( member_complex @ C @ A4 ) ) ).

% ComplD
thf(fact_6428_ComplD,axiom,
    ! [C: real,A4: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A4 ) )
     => ~ ( member_real @ C @ A4 ) ) ).

% ComplD
thf(fact_6429_ComplD,axiom,
    ! [C: $o,A4: set_o] :
      ( ( member_o @ C @ ( uminus_uminus_set_o @ A4 ) )
     => ~ ( member_o @ C @ A4 ) ) ).

% ComplD
thf(fact_6430_ComplD,axiom,
    ! [C: nat,A4: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A4 ) )
     => ~ ( member_nat @ C @ A4 ) ) ).

% ComplD
thf(fact_6431_ComplD,axiom,
    ! [C: int,A4: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A4 ) )
     => ~ ( member_int @ C @ A4 ) ) ).

% ComplD
thf(fact_6432_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X4: real] : ( member_real @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6433_uminus__set__def,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A5: set_o] :
          ( collect_o
          @ ( uminus_uminus_o_o
            @ ^ [X4: $o] : ( member_o @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6434_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X4: int] : ( member_int @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6435_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6436_uminus__set__def,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A5: set_complex] :
          ( collect_complex
          @ ( uminus1680532995456772888plex_o
            @ ^ [X4: complex] : ( member_complex @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6437_uminus__set__def,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( uminus8676089048583255045_nat_o
            @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6438_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X4: set_nat] : ( member_set_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6439_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X4: list_nat] : ( member_list_nat @ X4 @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6440_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ~ ( P @ X4 ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6441_Collect__neg__eq,axiom,
    ! [P: complex > $o] :
      ( ( collect_complex
        @ ^ [X4: complex] :
            ~ ( P @ X4 ) )
      = ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6442_Collect__neg__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] :
            ~ ( P @ X4 ) )
      = ( uminus6524753893492686040at_nat @ ( collec3392354462482085612at_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6443_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X4: set_nat] :
            ~ ( P @ X4 ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6444_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X4: list_nat] :
            ~ ( P @ X4 ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6445_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ^ [X4: real] :
              ~ ( member_real @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6446_Compl__eq,axiom,
    ( uminus_uminus_set_o
    = ( ^ [A5: set_o] :
          ( collect_o
          @ ^ [X4: $o] :
              ~ ( member_o @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6447_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ^ [X4: int] :
              ~ ( member_int @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6448_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ~ ( member_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6449_Compl__eq,axiom,
    ( uminus8566677241136511917omplex
    = ( ^ [A5: set_complex] :
          ( collect_complex
          @ ^ [X4: complex] :
              ~ ( member_complex @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6450_Compl__eq,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X4: product_prod_nat_nat] :
              ~ ( member8440522571783428010at_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6451_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] :
              ~ ( member_set_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6452_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X4: list_nat] :
              ~ ( member_list_nat @ X4 @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6453_mem__case__prodE,axiom,
    ! [Z2: complex,C: num > num > set_complex,P2: product_prod_num_num] :
      ( ( member_complex @ Z2 @ ( produc2866383454006189126omplex @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6454_mem__case__prodE,axiom,
    ! [Z2: real,C: num > num > set_real,P2: product_prod_num_num] :
      ( ( member_real @ Z2 @ ( produc8296048397933160132t_real @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_real @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6455_mem__case__prodE,axiom,
    ! [Z2: $o,C: num > num > set_o,P2: product_prod_num_num] :
      ( ( member_o @ Z2 @ ( produc1904233663560707350_set_o @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_o @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6456_mem__case__prodE,axiom,
    ! [Z2: nat,C: num > num > set_nat,P2: product_prod_num_num] :
      ( ( member_nat @ Z2 @ ( produc1361121860356118632et_nat @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6457_mem__case__prodE,axiom,
    ! [Z2: int,C: num > num > set_int,P2: product_prod_num_num] :
      ( ( member_int @ Z2 @ ( produc6406642877701697732et_int @ C @ P2 ) )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( member_int @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6458_mem__case__prodE,axiom,
    ! [Z2: complex,C: nat > num > set_complex,P2: product_prod_nat_num] :
      ( ( member_complex @ Z2 @ ( produc6231982587499038204omplex @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_complex @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6459_mem__case__prodE,axiom,
    ! [Z2: real,C: nat > num > set_real,P2: product_prod_nat_num] :
      ( ( member_real @ Z2 @ ( produc1435849484188172666t_real @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_real @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6460_mem__case__prodE,axiom,
    ! [Z2: $o,C: nat > num > set_o,P2: product_prod_nat_num] :
      ( ( member_o @ Z2 @ ( produc836176033315069408_set_o @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_o @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6461_mem__case__prodE,axiom,
    ! [Z2: nat,C: nat > num > set_nat,P2: product_prod_nat_num] :
      ( ( member_nat @ Z2 @ ( produc4130284055270567454et_nat @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_nat @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6462_mem__case__prodE,axiom,
    ! [Z2: int,C: nat > num > set_int,P2: product_prod_nat_num] :
      ( ( member_int @ Z2 @ ( produc9175805072616146554et_int @ C @ P2 ) )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( member_int @ Z2 @ ( C @ X5 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_6463_case__prodE,axiom,
    ! [C: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,P2: produc3843707927480180839at_nat] :
      ( ( produc410239310623530412_nat_o @ C @ P2 )
     => ~ ! [X5: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
            ( ( P2
              = ( produc2922128104949294807at_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6464_case__prodE,axiom,
    ! [C: num > num > $o,P2: product_prod_num_num] :
      ( ( produc5703948589228662326_num_o @ C @ P2 )
     => ~ ! [X5: num,Y4: num] :
            ( ( P2
              = ( product_Pair_num_num @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6465_case__prodE,axiom,
    ! [C: nat > num > $o,P2: product_prod_nat_num] :
      ( ( produc4927758841916487424_num_o @ C @ P2 )
     => ~ ! [X5: nat,Y4: num] :
            ( ( P2
              = ( product_Pair_nat_num @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6466_case__prodE,axiom,
    ! [C: int > int > $o,P2: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P2 )
     => ~ ! [X5: int,Y4: int] :
            ( ( P2
              = ( product_Pair_int_int @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6467_case__prodE,axiom,
    ! [C: nat > nat > $o,P2: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C @ P2 )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 ) ) ) ).

% case_prodE
thf(fact_6468_case__prodD,axiom,
    ! [F: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( produc410239310623530412_nat_o @ F @ ( produc2922128104949294807at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6469_case__prodD,axiom,
    ! [F: num > num > $o,A: num,B: num] :
      ( ( produc5703948589228662326_num_o @ F @ ( product_Pair_num_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6470_case__prodD,axiom,
    ! [F: nat > num > $o,A: nat,B: num] :
      ( ( produc4927758841916487424_num_o @ F @ ( product_Pair_nat_num @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6471_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6472_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_6473_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P2: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P2 @ Z2 )
     => ~ ! [X5: nat,Y4: nat] :
            ( ( P2
              = ( product_Pair_nat_nat @ X5 @ Y4 ) )
           => ~ ( C @ X5 @ Y4 @ Z2 ) ) ) ).

% case_prodE'
thf(fact_6474_case__prodD_H,axiom,
    ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_6475_Collect__case__prod__mono,axiom,
    ! [A4: nat > nat > $o,B4: nat > nat > $o] :
      ( ( ord_le2646555220125990790_nat_o @ A4 @ B4 )
     => ( ord_le3146513528884898305at_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ A4 ) ) @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ B4 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_6476_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6477_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_6478_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_int_def
thf(fact_6479_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_6480_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_6481_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_6482_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_def
thf(fact_6483_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M5 ) @ ( numeral_numeral_nat @ N3 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_6484_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6485_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X4: real] : ( minus_minus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6486_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X4: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6487_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6488_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K2 ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6489_take__bit__numeral__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_6490_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M5: num,N3: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N3 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique5026877609467782581ep_nat @ N3 @ ( unique5055182867167087721od_nat @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6491_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M5: num,N3: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M5 @ N3 ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M5 ) ) @ ( unique5024387138958732305ep_int @ N3 @ ( unique5052692396658037445od_int @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6492_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M5 @ N3 ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M5 ) ) @ ( unique4921790084139445826nteger @ N3 @ ( unique3479559517661332726nteger @ M5 @ ( bit0 @ N3 ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_6493_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).

% take_bit_numeral_bit1
thf(fact_6494_take__bit__numeral__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_bit1
thf(fact_6495_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_6496_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_6497_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_6498_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_6499_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M5 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M5 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M5 @ N3 ) @ N3 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6500_take__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_Suc_minus_bit1
thf(fact_6501_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_6502_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_6503_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_6504_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_6505_split__part,axiom,
    ! [P: $o,Q: nat > nat > $o] :
      ( ( produc6081775807080527818_nat_o
        @ ^ [A6: nat,B7: nat] :
            ( P
            & ( Q @ A6 @ B7 ) ) )
      = ( ^ [Ab: product_prod_nat_nat] :
            ( P
            & ( produc6081775807080527818_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_6506_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6507_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6508_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6509_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6510_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6511_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6512_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6513_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6514_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6515_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6516_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6517_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6518_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_6519_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_6520_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_6521_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_6522_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_6523_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_6524_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_6525_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_6526_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6527_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6528_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6529_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6530_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6531_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6532_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6533_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6534_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_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_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_6543_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_6544_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_6545_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_6546_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_6547_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_6548_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_6549_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_6550_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_6551_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_6552_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_6553_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_6554_pred__numeral__inc,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( inc @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% pred_numeral_inc
thf(fact_6555_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6556_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6557_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6558_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6559_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_6560_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6561_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_6562_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_6563_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R2 ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_6564_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_6565_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_6566_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_6567_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_6568_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_6569_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_6570_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_6571_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_6572_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_6573_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_6574_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_6575_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_6576_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_6577_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_6578_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_6579_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_6580_add__neg__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_6581_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_6582_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M2 ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_6583_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_6584_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_6585_add__neg__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M2 ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_6586_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_6587_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M2 ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( numeral_numeral_real @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_6588_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M2 ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( numera6690914467698888265omplex @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_6589_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_6590_diff__numeral__special_I6_J,axiom,
    ! [M2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M2 ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M2 ) ) ) ).

% diff_numeral_special(6)
thf(fact_6591_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_6592_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_6593_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_6594_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_6595_diff__numeral__special_I5_J,axiom,
    ! [N: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_6596_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_6597_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_6598_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_6599_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_6600_numeral__div__minus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_6601_minus__numeral__div__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_6602_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_nat_nat] :
      ( produc6081775807080527818_nat_o
      @ ^ [Uu3: nat,Uv3: nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_6603_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_6604_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_6605_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_6606_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_6607_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_6608_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_6609_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_6610_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_6611_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6612_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6613_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6614_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6615_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_6616_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_6617_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_6618_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_6619_num__induct,axiom,
    ! [P: num > $o,X3: num] :
      ( ( P @ one )
     => ( ! [X5: num] :
            ( ( P @ X5 )
           => ( P @ ( inc @ X5 ) ) )
       => ( P @ X3 ) ) ) ).

% num_induct
thf(fact_6620_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_6621_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_6622_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_6623_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_6624_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6625_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6626_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6627_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6628_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6629_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6630_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6631_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6632_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6633_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6634_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6635_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_6636_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6637_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6638_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6639_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_6640_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6641_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6642_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6643_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_6644_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6645_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6646_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6647_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_6648_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_6649_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6650_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_6651_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_6652_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6653_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6654_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6655_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_6656_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6657_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6658_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6659_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_6660_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6661_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6662_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6663_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_6664_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_6665_inc_Osimps_I2_J,axiom,
    ! [X3: num] :
      ( ( inc @ ( bit0 @ X3 ) )
      = ( bit1 @ X3 ) ) ).

% inc.simps(2)
thf(fact_6666_inc_Osimps_I3_J,axiom,
    ! [X3: num] :
      ( ( inc @ ( bit1 @ X3 ) )
      = ( bit0 @ ( inc @ X3 ) ) ) ).

% inc.simps(3)
thf(fact_6667_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_6668_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_6669_abs__mult__pos,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X3 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6670_abs__mult__pos,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X3 )
        = ( abs_abs_real @ ( times_times_real @ Y @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6671_abs__mult__pos,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X3 )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6672_abs__mult__pos,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X3 )
        = ( abs_abs_int @ ( times_times_int @ Y @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_6673_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6674_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6675_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6676_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_6677_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6678_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6679_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6680_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_6681_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6682_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6683_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6684_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_6685_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_6686_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6687_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_6688_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_6689_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_6690_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_6691_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_6692_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_6693_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_6694_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_6695_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6696_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_6697_abs__diff__le__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
        & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6698_abs__diff__le__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
        & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6699_abs__diff__le__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
        & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6700_abs__diff__le__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
        & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6701_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6702_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6703_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6704_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_6705_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6706_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6707_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6708_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_6709_abs__diff__less__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
        & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6710_abs__diff__less__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
        & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6711_abs__diff__less__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
        & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6712_abs__diff__less__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
        & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6713_add__One,axiom,
    ! [X3: num] :
      ( ( plus_plus_num @ X3 @ one )
      = ( inc @ X3 ) ) ).

% add_One
thf(fact_6714_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_6715_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_6716_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_6717_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_6718_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_6719_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_real @ ( inc @ X3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_6720_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_rat @ ( inc @ X3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_6721_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_nat @ ( inc @ X3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_6722_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_int @ ( inc @ X3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_6723_abs__le__square__iff,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6724_abs__le__square__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6725_abs__le__square__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6726_abs__le__square__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6727_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_6728_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_6729_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_6730_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_6731_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_6732_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_6733_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_6734_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_6735_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6736_power2__le__iff__abs__le,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6737_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6738_power2__le__iff__abs__le,axiom,
    ! [Y: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6739_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_6740_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_6741_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_6742_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_6743_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_6744_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_6745_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_6746_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_6747_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_6748_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_6749_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_6750_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_6751_power__mono__even,axiom,
    ! [N: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_6752_power__mono__even,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_6753_power__mono__even,axiom,
    ! [N: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_6754_power__mono__even,axiom,
    ! [N: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).

% power_mono_even
thf(fact_6755_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N3: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M5 @ N3 ) @ ( modulo_modulo_nat @ M5 @ N3 ) ) ) ) ).

% divmod_nat_def
thf(fact_6756_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_6757_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_6758_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_6759_and__int_Oelims,axiom,
    ! [X3: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y )
     => ( ( ( ( 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 ) ) ) )
         => ( Y
            = ( 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 ) ) ) )
         => ( Y
            = ( 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_6760_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
          @ ( uminus_uminus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_6761_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_6762_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6763_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6764_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6765_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_6766_divmod__BitM__2__eq,axiom,
    ! [M2: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M2 ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_6767_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6768_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6769_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6770_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_6771_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6772_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) )
      = ( numeral_numeral_real @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6773_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( numeral_numeral_rat @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6774_dbl__dec__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ ( bitM @ K2 ) ) ) ).

% dbl_dec_simps(5)
thf(fact_6775_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_6776_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_6777_and__numerals_I8_J,axiom,
    ! [X3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_6778_and__numerals_I8_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_6779_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit0 @ K2 ) )
      = ( numeral_numeral_nat @ ( bitM @ K2 ) ) ) ).

% pred_numeral_simps(2)
thf(fact_6780_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6781_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6782_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6783_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6784_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_6785_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6786_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6787_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6788_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6789_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_6790_and__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_6791_and__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_6792_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_6793_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_6794_and__numerals_I3_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6795_and__numerals_I3_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6796_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_6797_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_6798_and__numerals_I6_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6799_and__numerals_I6_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6800_and__numerals_I4_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6801_and__numerals_I4_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6802_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_6803_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_6804_and__numerals_I7_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6805_and__numerals_I7_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6806_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_6807_semiring__norm_I28_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ N ) ) ) ).

% semiring_norm(28)
thf(fact_6808_semiring__norm_I27_J,axiom,
    ! [N: num] :
      ( ( bitM @ ( bit0 @ N ) )
      = ( bit1 @ ( bitM @ N ) ) ) ).

% semiring_norm(27)
thf(fact_6809_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_6810_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_6811_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6812_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6813_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6814_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_6815_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_6816_even__and__iff__int,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K2 @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_6817_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_6818_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X4: real] : ( plus_plus_real @ ( plus_plus_real @ X4 @ X4 ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_6819_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X4: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X4 @ X4 ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_6820_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_6821_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6822_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6823_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6824_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6825_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6826_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6827_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_6828_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ ( bitM @ N ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_6829_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_6830_numeral__BitM,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ ( bitM @ N ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_6831_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_6832_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_6833_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_6834_even__abs__add__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K2 ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_abs_add_iff
thf(fact_6835_even__add__abs__iff,axiom,
    ! [K2: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K2 @ L ) ) ) ).

% even_add_abs_iff
thf(fact_6836_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F: nat > int,K2: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M2 @ 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 @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F @ M2 ) @ K2 )
         => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M2 @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F @ I3 )
                  = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6837_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_int_rec
thf(fact_6838_nat__ivt__aux,axiom,
    ! [N: nat,F: nat > int,K2: 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 ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K2 ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6839_nat0__intermed__int__val,axiom,
    ! [N: nat,F: nat > int,K2: 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 ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F @ I3 )
                = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6840_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_6841_and__int_Opsimps,axiom,
    ! [K2: int,L: int] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L ) )
     => ( ( ( ( member_int @ K2 @ ( 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 @ K2 @ L )
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
        & ( ~ ( ( member_int @ K2 @ ( 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 @ K2 @ L )
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.psimps
thf(fact_6842_and__int_Opelims,axiom,
    ! [X3: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y )
     => ( ( 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 ) ) ) )
               => ( Y
                  = ( 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 ) ) ) )
               => ( Y
                  = ( 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_6843_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6844_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6845_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6846_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6847_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6848_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N3 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N3 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_6849_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6850_mask__numeral,axiom,
    ! [N: num] :
      ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N ) ) ) ) ) ).

% mask_numeral
thf(fact_6851_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_6852_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = ( numera6690914467698888265omplex @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6853_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = ( numeral_numeral_real @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6854_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = ( numeral_numeral_rat @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6855_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = ( numeral_numeral_int @ N ) )
      = ( Z2
        = ( numeral_numeral_int @ N ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_6856_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K2 ) )
      = ( numera6690914467698888265omplex @ K2 ) ) ).

% of_int_numeral
thf(fact_6857_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_real @ K2 ) ) ).

% of_int_numeral
thf(fact_6858_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_rat @ K2 ) ) ).

% of_int_numeral
thf(fact_6859_of__int__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_int @ K2 ) ) ).

% of_int_numeral
thf(fact_6860_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ W @ Z2 ) ) ).

% of_int_le_iff
thf(fact_6861_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ W @ Z2 ) ) ).

% of_int_le_iff
thf(fact_6862_of__int__le__iff,axiom,
    ! [W: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ W @ Z2 ) ) ).

% of_int_le_iff
thf(fact_6863_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_add
thf(fact_6864_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_add
thf(fact_6865_of__int__add,axiom,
    ! [W: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z2 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_add
thf(fact_6866_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_6867_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_6868_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_6869_of__int__power,axiom,
    ! [Z2: int,N: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N ) ) ).

% of_int_power
thf(fact_6870_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X3 ) )
      = ( ( power_power_int @ B @ W )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6871_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X3 ) )
      = ( ( power_power_int @ B @ W )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6872_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X3 ) )
      = ( ( power_power_int @ B @ W )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6873_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X3 ) )
      = ( ( power_power_int @ B @ W )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6874_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X3 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X3
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6875_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X3 )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X3
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6876_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X3 )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X3
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6877_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X3 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X3
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6878_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_6879_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_6880_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M2 ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6881_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6882_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M2 ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6883_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6884_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6885_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6886_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6887_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6888_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6889_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6890_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6891_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6892_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6893_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6894_of__int__numeral__le__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6895_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6896_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6897_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6898_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_6899_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_6900_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6901_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6902_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6903_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6904_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6905_of__int__numeral__less__iff,axiom,
    ! [N: num,Z2: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6906_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6907_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6908_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6909_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6910_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6911_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6912_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6913_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6914_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6915_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6916_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6917_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6918_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6919_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6920_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6921_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6922_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6923_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6924_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6925_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6926_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6927_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6928_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X3: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6929_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6930_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6931_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6932_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_6933_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(2)
thf(fact_6934_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_6935_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_6936_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N ) ) ) ).

% bit_numeral_simps(3)
thf(fact_6937_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_6938_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6939_bit__0,axiom,
    ! [A: int] :
      ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6940_bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6941_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_6942_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_6943_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_6944_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_6945_bit__mod__2__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_6946_bit__mod__2__iff,axiom,
    ! [A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_6947_bit__mod__2__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N )
      = ( ( N = zero_zero_nat )
        & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_6948_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6949_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6950_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6951_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6952_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6953_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6954_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6955_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6956_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6957_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6958_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6959_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6960_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6961_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6962_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6963_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6964_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6965_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6966_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6967_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6968_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6969_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6970_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6971_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6972_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6973_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_6974_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6975_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6976_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6977_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_6978_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6979_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6980_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6981_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_6982_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6983_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6984_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6985_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_6986_bit__numeral__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M2 ) @ N )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% bit_numeral_iff
thf(fact_6987_bit__numeral__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% bit_numeral_iff
thf(fact_6988_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N )
        = ( ( bit_se1146084159140164899it_int @ A @ N )
          | ( bit_se1146084159140164899it_int @ B @ N ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_6989_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N )
        = ( ( bit_se1148574629649215175it_nat @ A @ N )
          | ( bit_se1148574629649215175it_nat @ B @ N ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_6990_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_6991_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6992_not__bit__1__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N ) ) ).

% not_bit_1_Suc
thf(fact_6993_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6994_bit__numeral__simps_I1_J,axiom,
    ! [N: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% bit_numeral_simps(1)
thf(fact_6995_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_6996_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6997_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6998_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6999_of__int__leD,axiom,
    ! [N: int,X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N ) ) @ X3 )
     => ( ( N = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X3 ) ) ) ).

% of_int_leD
thf(fact_7000_of__int__leD,axiom,
    ! [N: int,X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N ) ) @ X3 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% of_int_leD
thf(fact_7001_of__int__leD,axiom,
    ! [N: int,X3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N ) ) @ X3 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).

% of_int_leD
thf(fact_7002_of__int__leD,axiom,
    ! [N: int,X3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X3 )
     => ( ( N = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X3 ) ) ) ).

% of_int_leD
thf(fact_7003_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_7004_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_7005_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_7006_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_7007_of__int__neg__numeral,axiom,
    ! [K2: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).

% of_int_neg_numeral
thf(fact_7008_bit__concat__bit__iff,axiom,
    ! [M2: nat,K2: int,L: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M2 @ K2 @ L ) @ N )
      = ( ( ( ord_less_nat @ N @ M2 )
          & ( bit_se1146084159140164899it_int @ K2 @ N ) )
        | ( ( ord_less_eq_nat @ M2 @ N )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_7009_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N )
        = zero_zero_int )
     => ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_7010_exp__eq__0__imp__not__bit,axiom,
    ! [N: nat,A: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
        = zero_zero_nat )
     => ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ).

% exp_eq_0_imp_not_bit
thf(fact_7011_bit__Suc,axiom,
    ! [A: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_7012_bit__Suc,axiom,
    ! [A: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N ) )
      = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N ) ) ).

% bit_Suc
thf(fact_7013_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se9216721137139052372nteger @ A @ N )
        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_7014_stable__imp__bit__iff__odd,axiom,
    ! [A: int,N: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1146084159140164899it_int @ A @ N )
        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_7015_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,N: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N )
        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_7016_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N2: nat] :
          ( ( bit_se9216721137139052372nteger @ A @ N2 )
          = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_7017_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N2: nat] :
          ( ( bit_se1146084159140164899it_int @ A @ N2 )
          = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_7018_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N2: nat] :
          ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_7019_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N2: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N2 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ M3 )
                = ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N2 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N2 ) ) ) ) ) ).

% int_bit_bound
thf(fact_7020_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_7021_bit__iff__odd,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A6: code_integer,N3: nat] :
          ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A6 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_7022_bit__iff__odd,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A6: int,N3: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A6 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_7023_bit__iff__odd,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A6: nat,N3: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_iff_odd
thf(fact_7024_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: int,N: nat] :
      ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
        = zero_zero_int )
      = ( ~ ( bit_se1146084159140164899it_int @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_7025_and__exp__eq__0__iff__not__bit,axiom,
    ! [A: nat,N: nat] :
      ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
        = zero_zero_nat )
      = ( ~ ( bit_se1148574629649215175it_nat @ A @ N ) ) ) ).

% and_exp_eq_0_iff_not_bit
thf(fact_7026_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_7027_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_7028_even__of__int__iff,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ).

% even_of_int_iff
thf(fact_7029_even__of__int__iff,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K2 ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ).

% even_of_int_iff
thf(fact_7030_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N3: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_int_def
thf(fact_7031_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7032_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7033_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) )
      = ( N = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_7034_even__bit__succ__iff,axiom,
    ! [A: code_integer,N: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N )
        = ( ( bit_se9216721137139052372nteger @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7035_even__bit__succ__iff,axiom,
    ! [A: int,N: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N )
        = ( ( bit_se1146084159140164899it_int @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7036_even__bit__succ__iff,axiom,
    ! [A: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N )
        = ( ( bit_se1148574629649215175it_nat @ A @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_7037_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,N: nat] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ A @ N )
        = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_7038_odd__bit__iff__bit__pred,axiom,
    ! [A: int,N: nat] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1146084159140164899it_int @ A @ N )
        = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_7039_odd__bit__iff__bit__pred,axiom,
    ! [A: nat,N: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N )
        = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N )
          | ( N = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_7040_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_7041_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_7042_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_7043_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N3: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N3 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% set_bit_eq
thf(fact_7044_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N3: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N3 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_7045_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7046_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7047_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N )
        = ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_7048_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A6: code_integer,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A6 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A6 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_7049_bit__rec,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A6: int,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A6 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A6 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_7050_bit__rec,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A6: nat,N3: nat] :
          ( ( ( N3 = zero_zero_nat )
           => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A6 ) )
          & ( ( N3 != zero_zero_nat )
           => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_7051_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N3: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_7052_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N3: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_7053_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat
          @ ( ( M5 = zero_zero_nat )
            | ( N3 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7054_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7055_and__int_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [K: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L4 ) )
           => ( ( ~ ( ( member_int @ K @ ( 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 @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K @ L4 ) ) )
       => ( P @ A0 @ A1 ) ) ) ).

% and_int.pinduct
thf(fact_7056_take__bit__Suc__from__most,axiom,
    ! [N: nat,K2: int] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K2 )
      = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K2 @ N ) ) ) @ ( bit_se2923211474154528505it_int @ N @ K2 ) ) ) ).

% take_bit_Suc_from_most
thf(fact_7057_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N: nat,K2: int] :
      ( ( ( bit_se2923211474154528505it_int @ N @ K2 )
        = ( bit_se2000444600071755411sk_int @ N ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( plus_plus_int @ K2 @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_7058_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_7059_floor__exists1,axiom,
    ! [X3: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X3 )
            & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X5 ) ) ) ).

% floor_exists1
thf(fact_7060_floor__exists1,axiom,
    ! [X3: rat] :
    ? [X5: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X3 )
            & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X5 ) ) ) ).

% floor_exists1
thf(fact_7061_floor__exists,axiom,
    ! [X3: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7062_floor__exists,axiom,
    ! [X3: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7063_upto_Opinduct,axiom,
    ! [A0: int,A1: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
     => ( ! [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 @ A1 ) ) ) ).

% upto.pinduct
thf(fact_7064_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_7065_power__numeral,axiom,
    ! [K2: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K2 @ L ) ) ) ).

% power_numeral
thf(fact_7066_power__numeral,axiom,
    ! [K2: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K2 ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K2 @ L ) ) ) ).

% power_numeral
thf(fact_7067_power__numeral,axiom,
    ! [K2: num,L: num] :
      ( ( power_power_rat @ ( numeral_numeral_rat @ K2 ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_rat @ ( pow @ K2 @ L ) ) ) ).

% power_numeral
thf(fact_7068_power__numeral,axiom,
    ! [K2: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K2 ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K2 @ L ) ) ) ).

% power_numeral
thf(fact_7069_power__numeral,axiom,
    ! [K2: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K2 @ L ) ) ) ).

% power_numeral
thf(fact_7070_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_7071_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_7072_or__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(2)
thf(fact_7073_or__numerals_I8_J,axiom,
    ! [X3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X3 ) ) ) ).

% or_numerals(8)
thf(fact_7074_or__numerals_I8_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_numerals(8)
thf(fact_7075_or__numerals_I3_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_7076_or__numerals_I3_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% or_numerals(3)
thf(fact_7077_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_7078_or__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_numerals(1)
thf(fact_7079_or__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X3 ) ) ) ).

% or_numerals(5)
thf(fact_7080_or__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_numerals(5)
thf(fact_7081_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_7082_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_7083_or__numerals_I7_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7084_or__numerals_I7_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(7)
thf(fact_7085_or__numerals_I6_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7086_or__numerals_I6_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(6)
thf(fact_7087_or__numerals_I4_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7088_or__numerals_I4_J,axiom,
    ! [X3: num,Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% or_numerals(4)
thf(fact_7089_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_7090_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_7091_disjunctive__add,axiom,
    ! [A: int,B: int] :
      ( ! [N2: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N2 ) )
     => ( ( plus_plus_int @ A @ B )
        = ( bit_se1409905431419307370or_int @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_7092_disjunctive__add,axiom,
    ! [A: nat,B: nat] :
      ( ! [N2: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N2 ) )
     => ( ( plus_plus_nat @ A @ B )
        = ( bit_se1412395901928357646or_nat @ A @ B ) ) ) ).

% disjunctive_add
thf(fact_7093_exp__add__commuting,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( times_times_complex @ X3 @ Y )
        = ( times_times_complex @ Y @ X3 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7094_exp__add__commuting,axiom,
    ! [X3: real,Y: real] :
      ( ( ( times_times_real @ X3 @ Y )
        = ( times_times_real @ Y @ X3 ) )
     => ( ( exp_real @ ( plus_plus_real @ X3 @ Y ) )
        = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_7095_mult__exp__exp,axiom,
    ! [X3: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X3 @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7096_mult__exp__exp,axiom,
    ! [X3: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X3 @ Y ) ) ) ).

% mult_exp_exp
thf(fact_7097_pow_Osimps_I1_J,axiom,
    ! [X3: num] :
      ( ( pow @ X3 @ one )
      = X3 ) ).

% pow.simps(1)
thf(fact_7098_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_7099_even__or__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1080825931792720795nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_7100_even__or__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_7101_even__or__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_or_iff
thf(fact_7102_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_7103_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7104_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_7105_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M5: nat,N3: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ) ).

% bit_nat_def
thf(fact_7106_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_7107_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ).

% mask_Suc_exp
thf(fact_7108_mask__Suc__exp,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ).

% mask_Suc_exp
thf(fact_7109_exp__double,axiom,
    ! [Z2: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) )
      = ( power_power_complex @ ( exp_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7110_exp__double,axiom,
    ! [Z2: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) )
      = ( power_power_real @ ( exp_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_7111_one__or__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ one_one_Code_integer @ A )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_7112_one__or__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ A )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_7113_one__or__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ one_one_nat @ A )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_or_eq
thf(fact_7114_or__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se1080825931792720795nteger @ A @ one_one_Code_integer )
      = ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_7115_or__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se1409905431419307370or_int @ A @ one_one_int )
      = ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_7116_or__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se1412395901928357646or_nat @ A @ one_one_nat )
      = ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% or_one_eq
thf(fact_7117_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
      = ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_7118_mask__Suc__double,axiom,
    ! [N: nat] :
      ( ( bit_se2000444600071755411sk_int @ ( suc @ N ) )
      = ( bit_se1409905431419307370or_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% mask_Suc_double
thf(fact_7119_OR__upper,axiom,
    ! [X3: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% OR_upper
thf(fact_7120_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N2: nat] :
            ( ~ ( P @ N2 )
            & ( P @ ( suc @ N2 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_7121_ex__le__of__int,axiom,
    ! [X3: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_7122_ex__le__of__int,axiom,
    ! [X3: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_7123_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_7124_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
              | ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_7125_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_7126_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_7127_round__unique,axiom,
    ! [X3: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X3 )
          = Y ) ) ) ).

% round_unique
thf(fact_7128_round__unique,axiom,
    ! [X3: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X3 )
          = Y ) ) ) ).

% round_unique
thf(fact_7129_round__unique_H,axiom,
    ! [X3: real,N: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ N ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X3 )
        = N ) ) ).

% round_unique'
thf(fact_7130_round__unique_H,axiom,
    ! [X3: rat,N: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ N ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X3 )
        = N ) ) ).

% round_unique'
thf(fact_7131_of__int__round__abs__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ X3 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7132_of__int__round__abs__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ X3 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7133_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_7134_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_7135_of__int__round__gt,axiom,
    ! [X3: real] : ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).

% of_int_round_gt
thf(fact_7136_of__int__round__gt,axiom,
    ! [X3: rat] : ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).

% of_int_round_gt
thf(fact_7137_round__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_7138_round__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% round_numeral
thf(fact_7139_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_7140_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_7141_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_7142_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_7143_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_7144_round__neg__numeral,axiom,
    ! [N: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% round_neg_numeral
thf(fact_7145_or__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_7146_or__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_7147_or__minus__numerals_I7_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_7148_or__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_7149_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_7150_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_7151_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M2 ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_7152_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M2: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M2 ) )
      = ( bit1 @ M2 ) ) ).

% or_not_num_neg.simps(3)
thf(fact_7153_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_7154_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M2 ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_7155_round__mono,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X3 ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_7156_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M2: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M2 ) )
      = ( bit1 @ M2 ) ) ).

% or_not_num_neg.simps(2)
thf(fact_7157_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M2 ) )
      = ( bitM @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_7158_or__not__num__neg_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != one ) ) )
       => ( ( ( X3 = one )
           => ! [M: num] :
                ( ( Xa2
                  = ( bit0 @ M ) )
               => ( Y
                 != ( bit1 @ M ) ) ) )
         => ( ( ( X3 = one )
             => ! [M: num] :
                  ( ( Xa2
                    = ( bit1 @ M ) )
                 => ( Y
                   != ( bit1 @ M ) ) ) )
           => ( ( ? [N2: num] :
                    ( X3
                    = ( bit0 @ N2 ) )
               => ( ( Xa2 = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N2: num] :
                    ( ( X3
                      = ( bit0 @ N2 ) )
                   => ! [M: num] :
                        ( ( Xa2
                          = ( bit0 @ M ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X3
                        = ( bit0 @ N2 ) )
                     => ! [M: num] :
                          ( ( Xa2
                            = ( bit1 @ M ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
                 => ( ( ? [N2: num] :
                          ( X3
                          = ( bit1 @ N2 ) )
                     => ( ( Xa2 = one )
                       => ( Y != one ) ) )
                   => ( ! [N2: num] :
                          ( ( X3
                            = ( bit1 @ N2 ) )
                         => ! [M: num] :
                              ( ( Xa2
                                = ( bit0 @ M ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) )
                     => ~ ! [N2: num] :
                            ( ( X3
                              = ( bit1 @ N2 ) )
                           => ! [M: num] :
                                ( ( Xa2
                                  = ( bit1 @ M ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_7159_round__diff__minimal,axiom,
    ! [Z2: real,M2: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_7160_round__diff__minimal,axiom,
    ! [Z2: rat,M2: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M2 ) ) ) ) ).

% round_diff_minimal
thf(fact_7161_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_7162_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_7163_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_7164_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_7165_of__int__round__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7166_of__int__round__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7167_of__int__round__ge,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).

% of_int_round_ge
thf(fact_7168_of__int__round__ge,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).

% of_int_round_ge
thf(fact_7169_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_7170_arctan__half,axiom,
    ( arctan
    = ( ^ [X4: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X4 @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_7171_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_7172_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_7173_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_7174_sqrt__sum__squares__half__less,axiom,
    ! [X3: real,U: real,Y: real] :
      ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_7175_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_7176_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_7177_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_7178_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_7179_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X3: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7180_real__sqrt__power,axiom,
    ! [X3: real,K2: nat] :
      ( ( sqrt @ ( power_power_real @ X3 @ K2 ) )
      = ( power_power_real @ ( sqrt @ X3 ) @ K2 ) ) ).

% real_sqrt_power
thf(fact_7181_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_7182_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7183_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7184_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_7185_real__less__rsqrt,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X3 @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_7186_sqrt__le__D,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y )
     => ( ord_less_eq_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_7187_real__le__rsqrt,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X3 @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_7188_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7189_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_7190_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_7191_real__le__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_7192_real__sqrt__unique,axiom,
    ! [Y: real,X3: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X3 )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_7193_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_7194_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X3: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X3 )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_7195_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X3: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X3 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_7196_real__sqrt__sum__squares__ge1,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_7197_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X3: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_7198_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_7199_sqrt__ge__absD,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_7200_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_7201_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_7202_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_7203_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7204_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7205_real__less__lsqrt,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X3 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X3 ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_7206_sqrt__sum__squares__le__sum,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X3 @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_7207_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_7208_real__sqrt__ge__abs1,axiom,
    ! [X3: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_7209_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_7210_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_7211_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_7212_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_7213_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_7214_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_7215_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_7216_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_7217_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X3: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7218_arith__geo__mean__sqrt,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X3 @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_7219_cos__x__y__le__one,axiom,
    ! [X3: real,Y: 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 @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_7220_real__sqrt__sum__squares__less,axiom,
    ! [X3: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_7221_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_7222_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_7223_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_7224_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_7225_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_7226_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_7227_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_7228_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7229_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7230_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_7231_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_7232_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_7233_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_7234_le__arcsin__iff,axiom,
    ! [X3: real,Y: 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 ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X3 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X3 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_7235_arcsin__le__iff,axiom,
    ! [X3: real,Y: 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 ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X3 ) @ Y )
              = ( ord_less_eq_real @ X3 @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_7236_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_7237_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_7238_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_7239_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_7240_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_7241_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_7242_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_7243_sin__monotone__2pi__le,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X3 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_7244_sin__mono__le__eq,axiom,
    ! [X3: real,Y: 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 ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_7245_sin__inj__pi,axiom,
    ! [X3: real,Y: 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 ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X3 )
                = ( sin_real @ Y ) )
             => ( X3 = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_7246_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_7247_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_7248_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_7249_sin__mono__less__eq,axiom,
    ! [X3: real,Y: 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 ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X3 @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_7250_sin__monotone__2pi,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X3 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_7251_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
            & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X5 )
              = Y )
            & ! [Y6: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
                  & ( ord_less_eq_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y6 )
                    = Y ) )
               => ( Y6 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_7252_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_7253_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_7254_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_7255_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_7256_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_7257_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_7258_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_7259_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_7260_sincos__total__2pi,axiom,
    ! [X3: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
           => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X3
                  = ( cos_real @ T5 ) )
               => ( Y
                 != ( sin_real @ T5 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7261_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_7262_sincos__total__2pi__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T5 )
          & ( ord_less_eq_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X3
            = ( cos_real @ T5 ) )
          & ( Y
            = ( sin_real @ T5 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7263_sincos__total__pi__half,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T5: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T5 )
              & ( ord_less_eq_real @ T5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X3
                = ( cos_real @ T5 ) )
              & ( Y
                = ( sin_real @ T5 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7264_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_7265_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_7266_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_7267_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_7268_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_7269_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_7270_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_7271_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_7272_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri681578069525770553at_rat @ M2 )
        = zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7273_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7274_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7275_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7276_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_7277_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_7278_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_7279_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_7280_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_7281_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_7282_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_7283_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_7284_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_7285_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_7286_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_7287_of__nat__less__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_iff
thf(fact_7288_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_7289_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri681578069525770553at_rat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% of_nat_numeral
thf(fact_7290_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_7291_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_7292_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_7293_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_7294_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_7295_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_7296_of__nat__le__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% of_nat_le_iff
thf(fact_7297_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_add
thf(fact_7298_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_add
thf(fact_7299_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_add
thf(fact_7300_of__nat__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_add
thf(fact_7301_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% of_nat_mult
thf(fact_7302_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% of_nat_mult
thf(fact_7303_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% of_nat_mult
thf(fact_7304_of__nat__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M2 @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% of_nat_mult
thf(fact_7305_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7306_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_7307_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_7308_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_7309_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_7310_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7311_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_7312_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_7313_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_7314_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_7315_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_7316_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_7317_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_7318_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_7319_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_7320_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_7321_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_7322_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_7323_of__nat__power,axiom,
    ! [M2: nat,N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N ) ) ).

% of_nat_power
thf(fact_7324_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X3 ) )
      = ( ( power_power_nat @ B @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7325_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X3 ) )
      = ( ( power_power_nat @ B @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7326_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X3 ) )
      = ( ( power_power_nat @ B @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7327_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ( power_power_nat @ B @ W )
        = X3 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_7328_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X3 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X3
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7329_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X3 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X3
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7330_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X3 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X3
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7331_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X3 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X3
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_7332_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_7333_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_7334_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_7335_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_7336_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7337_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7338_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7339_of__nat__le__0__iff,axiom,
    ! [M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7340_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_7341_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_7342_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_7343_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_7344_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_7345_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_7346_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_7347_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_7348_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_7349_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7350_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7351_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7352_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7353_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7354_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7355_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7356_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7357_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7358_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: nat] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N )
        = ( semiri681578069525770553at_rat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7359_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7360_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7361_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_7362_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,N: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7363_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,N: nat] :
      ( ( ( semiri681578069525770553at_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7364_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7365_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7366_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X3: num,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_7367_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7368_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7369_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7370_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7371_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7372_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7373_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7374_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7375_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_7376_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_7377_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_7378_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_7379_numeral__le__real__of__nat__iff,axiom,
    ! [N: num,M2: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ ( semiri5074537144036343181t_real @ M2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ M2 ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_7380_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_7381_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_7382_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_7383_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_7384_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_7385_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_7386_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_7387_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_7388_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_7389_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_7390_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_7391_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_7392_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_7393_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_7394_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_7395_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_7396_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_7397_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_7398_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_7399_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_7400_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_7401_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_7402_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_7403_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_7404_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_7405_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7406_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7407_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_7408_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_7409_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_7410_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_7411_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_7412_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_7413_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_7414_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_7415_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_7416_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_7417_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_7418_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_7419_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_7420_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_7421_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_7422_cos__pi__eq__zero,axiom,
    ! [M2: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_7423_real__arch__simple,axiom,
    ! [X3: real] :
    ? [N2: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ).

% real_arch_simple
thf(fact_7424_real__arch__simple,axiom,
    ! [X3: rat] :
    ? [N2: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% real_arch_simple
thf(fact_7425_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7426_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7427_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7428_mult__of__nat__commute,axiom,
    ! [X3: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7429_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_7430_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_7431_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_7432_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_7433_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_7434_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_7435_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_7436_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_7437_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_7438_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_7439_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_7440_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_7441_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_7442_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_7443_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_7444_of__nat__less__imp__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% of_nat_less_imp_less
thf(fact_7445_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_7446_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_7447_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_7448_less__imp__of__nat__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% less_imp_of_nat_less
thf(fact_7449_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_7450_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_7451_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_7452_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_7453_of__nat__max,axiom,
    ! [X3: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X3 @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_7454_of__nat__max,axiom,
    ! [X3: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X3 @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_7455_of__nat__max,axiom,
    ! [X3: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X3 @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_7456_add__tan__eq,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7457_add__tan__eq,axiom,
    ! [X3: real,Y: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7458_sin__add,axiom,
    ! [X3: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_7459_lemma__tan__add1,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7460_lemma__tan__add1,axiom,
    ! [X3: real,Y: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7461_tan__diff,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7462_tan__diff,axiom,
    ! [X3: real,Y: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X3 @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7463_tan__add,axiom,
    ! [X3: complex,Y: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7464_tan__add,axiom,
    ! [X3: real,Y: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X3 @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7465_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri681578069525770553at_rat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_7466_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% of_nat_diff
thf(fact_7467_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).

% of_nat_diff
thf(fact_7468_of__nat__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).

% of_nat_diff
thf(fact_7469_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_7470_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_7471_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_7472_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_7473_cos__one__2pi,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = one_one_real )
      = ( ? [X4: nat] :
            ( X3
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X4: nat] :
            ( X3
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7474_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_7475_cos__add,axiom,
    ! [X3: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_7476_cos__diff,axiom,
    ! [X3: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X3 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_7477_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7478_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( semiri4939895301339042750nteger @ N ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M2 ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) @ ( semiri4939895301339042750nteger @ N ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7479_mod__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7480_mod__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7481_tan__half,axiom,
    ( tan_complex
    = ( ^ [X4: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_7482_tan__half,axiom,
    ( tan_real
    = ( ^ [X4: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_7483_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N3: nat,M5: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M5 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7484_less__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N ) @ M2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).

% less_log_of_power
thf(fact_7485_log__of__power__eq,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ) ).

% log_of_power_eq
thf(fact_7486_sin__expansion__lemma,axiom,
    ! [X3: real,M2: nat] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7487_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_7488_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_7489_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N2: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7490_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N2: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_7491_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_7492_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_7493_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_7494_cos__zero__lemma,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( 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 ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_7495_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7496_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 )
      & ! [Y6: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y6 )
            & ( ord_less_eq_real @ Y6 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y6 )
              = zero_zero_real ) )
         => ( Y6 = X5 ) ) ) ).

% cos_is_zero
thf(fact_7497_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_7498_cos__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( 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 ) ) ) ) ) )
        | ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X3
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7499_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7500_inverse__of__nat__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( N != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7501_cos__expansion__lemma,axiom,
    ! [X3: real,M2: nat] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7502_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_7503_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_7504_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_7505_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_7506_le__log__of__power,axiom,
    ! [B: real,N: nat,M2: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N ) @ M2 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ M2 ) ) ) ) ).

% le_log_of_power
thf(fact_7507_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_7508_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_7509_log__nat__power,axiom,
    ! [X3: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ B @ ( power_power_real @ X3 @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ B @ X3 ) ) ) ) ).

% log_nat_power
thf(fact_7510_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_7511_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7512_cos__one__2pi__int,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = one_one_real )
      = ( ? [X4: int] :
            ( X3
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_7513_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_7514_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_7515_log2__of__power__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( semiri5074537144036343181t_real @ N )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% log2_of_power_eq
thf(fact_7516_log__of__power__less,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_7517_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_7518_cos__plus__cos,axiom,
    ! [W: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7519_cos__plus__cos,axiom,
    ! [W: real,Z2: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7520_cos__times__cos,axiom,
    ! [W: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7521_cos__times__cos,axiom,
    ! [W: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7522_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_7523_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_7524_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_7525_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_7526_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X5 ) ) ) ) ).

% lemma_tan_total
thf(fact_7527_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_7528_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_7529_tan__mono__lt__eq,axiom,
    ! [X3: real,Y: 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 ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X3 @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_7530_tan__monotone_H,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X3 )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_7531_tan__monotone,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X3 )
       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X3 ) ) ) ) ) ).

% tan_monotone
thf(fact_7532_tan__total,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y )
      & ! [Y6: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
            & ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y6 )
              = Y ) )
         => ( Y6 = X5 ) ) ) ).

% tan_total
thf(fact_7533_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_7534_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_7535_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_7536_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_7537_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_7538_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_7539_log__of__power__le,axiom,
    ! [M2: nat,B: real,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( power_power_real @ B @ N ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M2 )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_7540_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_7541_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_7542_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_7543_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_7544_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_7545_cos__diff__cos,axiom,
    ! [W: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7546_cos__diff__cos,axiom,
    ! [W: real,Z2: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7547_sin__diff__sin,axiom,
    ! [W: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7548_sin__diff__sin,axiom,
    ! [W: real,Z2: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7549_sin__plus__sin,axiom,
    ! [W: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7550_sin__plus__sin,axiom,
    ! [W: real,Z2: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7551_cos__times__sin,axiom,
    ! [W: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7552_cos__times__sin,axiom,
    ! [W: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7553_sin__times__cos,axiom,
    ! [W: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7554_sin__times__cos,axiom,
    ! [W: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7555_sin__times__sin,axiom,
    ! [W: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7556_sin__times__sin,axiom,
    ! [W: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7557_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_7558_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_7559_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_7560_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X5 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_7561_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_7562_tan__mono__le__eq,axiom,
    ! [X3: real,Y: 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 ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X3 @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_7563_tan__mono__le,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_7564_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_7565_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_7566_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_7567_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_7568_arctan__unique,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X3 )
            = Y )
         => ( ( arctan @ Y )
            = X3 ) ) ) ) ).

% arctan_unique
thf(fact_7569_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_7570_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_7571_less__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% less_log2_of_power
thf(fact_7572_le__log2__of__power,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ M2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% le_log2_of_power
thf(fact_7573_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_7574_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_7575_tan__total__pi4,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ? [Z3: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
          & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z3 )
            = X3 ) ) ) ).

% tan_total_pi4
thf(fact_7576_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_7577_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_7578_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_7579_log2__of__power__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_7580_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_7581_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_7582_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_7583_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_7584_sincos__total__pi,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
            & ( ord_less_eq_real @ T5 @ pi )
            & ( X3
              = ( cos_real @ T5 ) )
            & ( Y
              = ( sin_real @ T5 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7585_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_7586_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_7587_log2__of__power__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_7588_sin__zero__lemma,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( 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 ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_7589_sin__zero__iff,axiom,
    ! [X3: real] :
      ( ( ( 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 ) ) ) ) ) )
        | ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X3
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7590_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_7591_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M5: nat,Q4: nat] : ( if_complex @ ( Q4 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M5 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7592_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M5: nat,Q4: nat] : ( if_rat @ ( Q4 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M5 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7593_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M5: nat,Q4: nat] : ( if_real @ ( Q4 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M5 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7594_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M5: nat,Q4: nat] : ( if_int @ ( Q4 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M5 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7595_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M5: nat,Q4: nat] : ( if_nat @ ( Q4 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M5 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7596_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_7597_monoseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7598_lemma__termdiff3,axiom,
    ! [H: real,Z2: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( 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 @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7599_lemma__termdiff3,axiom,
    ! [H: complex,Z2: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( 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 @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_7600_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
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N3 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7601_arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( arctan @ X3 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7602_int__eq__iff__numeral,axiom,
    ! [M2: nat,V2: num] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( numeral_numeral_int @ V2 ) )
      = ( M2
        = ( numeral_numeral_nat @ V2 ) ) ) ).

% int_eq_iff_numeral
thf(fact_7603_negative__zless,axiom,
    ! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zless
thf(fact_7604_powser__zero,axiom,
    ! [F: nat > complex] :
      ( ( suminf_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7605_powser__zero,axiom,
    ! [F: nat > real] :
      ( ( suminf_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) )
      = ( F @ zero_zero_nat ) ) ).

% powser_zero
thf(fact_7606_int__ops_I3_J,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% int_ops(3)
thf(fact_7607_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_7608_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N2: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N2 ) )
     => ~ ! [N2: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% int_cases
thf(fact_7609_zle__int,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% zle_int
thf(fact_7610_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_7611_zadd__int__left,axiom,
    ! [M2: nat,N: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_7612_int__plus,axiom,
    ! [N: nat,M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% int_plus
thf(fact_7613_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_7614_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B7: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ).

% nat_leq_as_int
thf(fact_7615_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7616_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_7617_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z4: int] :
        ? [N3: nat] :
          ( Z4
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_7618_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_7619_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_7620_negD,axiom,
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ zero_zero_int )
     => ? [N2: nat] :
          ( X3
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).

% negD
thf(fact_7621_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7622_zdiff__int__split,axiom,
    ! [P: int > $o,X3: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X3 @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_7623_exp__bound__half,axiom,
    ! [Z2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7624_exp__bound__half,axiom,
    ! [Z2: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_7625_exp__bound__lemma,axiom,
    ! [Z2: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7626_exp__bound__lemma,axiom,
    ! [Z2: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z2 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_7627_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_7628_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_7629_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_7630_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_7631_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_7632_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_7633_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_7634_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_7635_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_7636_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_7637_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_7638_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7639_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_7640_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_7641_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_7642_norm__uminus__minus,axiom,
    ! [X3: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7643_norm__uminus__minus,axiom,
    ! [X3: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_7644_power__eq__imp__eq__norm,axiom,
    ! [W: real,N: nat,Z2: real] :
      ( ( ( power_power_real @ W @ N )
        = ( power_power_real @ Z2 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7645_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N: nat,Z2: complex] :
      ( ( ( power_power_complex @ W @ N )
        = ( power_power_complex @ Z2 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7646_norm__triangle__lt,axiom,
    ! [X3: real,Y: real,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7647_norm__triangle__lt,axiom,
    ! [X3: complex,Y: complex,E: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ E ) ) ).

% norm_triangle_lt
thf(fact_7648_norm__add__less,axiom,
    ! [X3: real,R2: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7649_norm__add__less,axiom,
    ! [X3: complex,R2: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7650_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_7651_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_7652_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7653_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7654_norm__triangle__ineq,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_7655_norm__triangle__ineq,axiom,
    ! [X3: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_7656_norm__triangle__le,axiom,
    ! [X3: real,Y: real,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7657_norm__triangle__le,axiom,
    ! [X3: complex,Y: complex,E: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y ) ) @ E ) ) ).

% norm_triangle_le
thf(fact_7658_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7659_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_7660_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_7661_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_7662_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_7663_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_7664_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7665_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_7666_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_7667_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_7668_norm__power__diff,axiom,
    ! [Z2: real,W: real,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ 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 @ Z2 @ M2 ) @ ( power_power_real @ W @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z2 @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7669_norm__power__diff,axiom,
    ! [Z2: complex,W: complex,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ 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 @ Z2 @ M2 ) @ ( power_power_complex @ W @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z2 @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_7670_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_7671_summable__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_7672_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N ) @ K2 )
     => ( ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_7673_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_7674_sum__gp,axiom,
    ! [N: nat,M2: nat,X3: complex] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X3 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X3 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M2 ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7675_sum__gp,axiom,
    ! [N: nat,M2: nat,X3: rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X3 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X3 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M2 ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7676_sum__gp,axiom,
    ! [N: nat,M2: nat,X3: real] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X3 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X3 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M2 ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7677_summable__iff__shift,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
      = ( summable_real @ F ) ) ).

% summable_iff_shift
thf(fact_7678_sum_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups8691415230153176458o_real @ G @ bot_bot_set_o )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7679_sum_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups7872700643590313910_o_rat @ G @ bot_bot_set_o )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7680_sum_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups8507830703676809646_o_nat @ G @ bot_bot_set_o )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7681_sum_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups8505340233167759370_o_int @ G @ bot_bot_set_o )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7682_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7683_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_7684_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_7685_sum_Oempty,axiom,
    ! [G: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_7686_sum_Oempty,axiom,
    ! [G: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7687_sum_Oempty,axiom,
    ! [G: nat > nat] :
      ( ( groups3542108847815614940at_nat @ G @ bot_bot_set_nat )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_7688_ceiling__numeral,axiom,
    ! [V2: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V2 ) )
      = ( numeral_numeral_int @ V2 ) ) ).

% ceiling_numeral
thf(fact_7689_ceiling__numeral,axiom,
    ! [V2: num] :
      ( ( archim2889992004027027881ng_rat @ ( numeral_numeral_rat @ V2 ) )
      = ( numeral_numeral_int @ V2 ) ) ).

% ceiling_numeral
thf(fact_7690_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_7691_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_7692_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X3 @ A4 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A4 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7693_sum_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X3 @ A4 )
       => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X3 @ A4 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7694_sum_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X3 @ A4 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A4 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7695_sum_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X3 @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A4 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7696_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X3 @ A4 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A4 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7697_sum_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X3 @ A4 )
       => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7698_sum_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X3 @ A4 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7699_sum_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X3 @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7700_sum_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X3 @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7701_sum_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X3 @ A4 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A4 ) )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ).

% sum.insert
thf(fact_7702_ceiling__add__of__int,axiom,
    ! [X3: rat,Z2: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 ) ) ).

% ceiling_add_of_int
thf(fact_7703_ceiling__add__of__int,axiom,
    ! [X3: real,Z2: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 ) ) ).

% ceiling_add_of_int
thf(fact_7704_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_7705_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_7706_ceiling__le__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_7707_ceiling__le__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_7708_ceiling__le__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ V2 ) ) ) ).

% ceiling_le_numeral
thf(fact_7709_ceiling__le__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_eq_rat @ X3 @ ( numeral_numeral_rat @ V2 ) ) ) ).

% ceiling_le_numeral
thf(fact_7710_ceiling__less__one,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_7711_ceiling__less__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
      = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_7712_numeral__less__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V2 ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V2 ) @ X3 ) ) ).

% numeral_less_ceiling
thf(fact_7713_numeral__less__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V2 ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V2 ) @ X3 ) ) ).

% numeral_less_ceiling
thf(fact_7714_ceiling__le__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
      = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_7715_ceiling__le__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
      = ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_7716_ceiling__add__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ V2 ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_add_numeral
thf(fact_7717_ceiling__add__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( numeral_numeral_rat @ V2 ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_add_numeral
thf(fact_7718_ceiling__neg__numeral,axiom,
    ! [V2: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_neg_numeral
thf(fact_7719_ceiling__neg__numeral,axiom,
    ! [V2: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_neg_numeral
thf(fact_7720_ceiling__add__one,axiom,
    ! [X3: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7721_ceiling__add__one,axiom,
    ! [X3: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7722_ceiling__diff__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X3 @ ( numeral_numeral_real @ V2 ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_diff_numeral
thf(fact_7723_ceiling__diff__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X3 @ ( numeral_numeral_rat @ V2 ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% ceiling_diff_numeral
thf(fact_7724_ceiling__numeral__power,axiom,
    ! [X3: num,N: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_7725_ceiling__numeral__power,axiom,
    ! [X3: num,N: nat] :
      ( ( archim2889992004027027881ng_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ).

% ceiling_numeral_power
thf(fact_7726_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_7727_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_7728_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7729_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7730_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7731_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_7732_ceiling__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_7733_ceiling__less__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_7734_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_7735_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_7736_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_7737_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7738_ceiling__less__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( numeral_numeral_real @ V2 ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_7739_ceiling__less__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V2 ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_7740_numeral__le__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V2 ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V2 ) @ one_one_real ) @ X3 ) ) ).

% numeral_le_ceiling
thf(fact_7741_numeral__le__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V2 ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V2 ) @ one_one_rat ) @ X3 ) ) ).

% numeral_le_ceiling
thf(fact_7742_ceiling__le__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7743_ceiling__le__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7744_neg__numeral__less__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ X3 ) ) ).

% neg_numeral_less_ceiling
thf(fact_7745_neg__numeral__less__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ X3 ) ) ).

% neg_numeral_less_ceiling
thf(fact_7746_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_7747_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_7748_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_7749_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7750_ceiling__less__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7751_ceiling__less__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7752_neg__numeral__le__ceiling,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ one_one_real ) @ X3 ) ) ).

% neg_numeral_le_ceiling
thf(fact_7753_neg__numeral__le__ceiling,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ one_one_rat ) @ X3 ) ) ).

% neg_numeral_le_ceiling
thf(fact_7754_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_7755_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_7756_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7757_summable__comparison__test_H,axiom,
    ! [G: nat > real,N5: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_7758_summable__add,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ( summable_real @ F )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7759_summable__add,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ( summable_nat @ F )
     => ( ( summable_nat @ G )
       => ( summable_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7760_summable__add,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ( summable_int @ F )
     => ( ( summable_int @ G )
       => ( summable_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ).

% summable_add
thf(fact_7761_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_7762_summable__ignore__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) ) ) ).

% summable_ignore_initial_segment
thf(fact_7763_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_7764_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_7765_sum__mono,axiom,
    ! [K5: set_o,F: $o > rat,G: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ K5 ) @ ( groups7872700643590313910_o_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7766_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_7767_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_7768_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_7769_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_7770_sum__mono,axiom,
    ! [K5: set_o,F: $o > nat,G: $o > nat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ K5 ) @ ( groups8507830703676809646_o_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7771_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_7772_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_7773_sum_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( plus_plus_nat @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7774_sum_Odistrib,axiom,
    ! [G: complex > complex,H: complex > complex,A4: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X4: complex] : ( plus_plus_complex @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7775_sum_Odistrib,axiom,
    ! [G: nat > real,H: nat > real,A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X4: nat] : ( plus_plus_real @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7776_sum_Odistrib,axiom,
    ! [G: int > int,H: int > int,A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X4: int] : ( plus_plus_int @ ( G @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ H @ A4 ) ) ) ).

% sum.distrib
thf(fact_7777_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7778_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7779_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N2: nat] :
              ( ( member_nat @ N2 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_7780_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_7781_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_7782_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7783_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_7784_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_7785_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_7786_sum__nonpos,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7787_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_7788_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_7789_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_7790_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_7791_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_7792_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7793_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_7794_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_7795_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_7796_sum__nonneg,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) ) ) ).

% sum_nonneg
thf(fact_7797_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_7798_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_7799_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_7800_powser__insidea,axiom,
    ! [F: nat > real,X3: real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7801_powser__insidea,axiom,
    ! [F: nat > complex,X3: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ) ) ).

% powser_insidea
thf(fact_7802_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_7803_sum__mono__inv,axiom,
    ! [F: $o > rat,I5: set_o,G: $o > rat,I: $o] :
      ( ( ( groups7872700643590313910_o_rat @ F @ I5 )
        = ( groups7872700643590313910_o_rat @ G @ I5 ) )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7804_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_7805_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_7806_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_7807_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_7808_sum__mono__inv,axiom,
    ! [F: $o > nat,I5: set_o,G: $o > nat,I: $o] :
      ( ( ( groups8507830703676809646_o_nat @ F @ I5 )
        = ( groups8507830703676809646_o_nat @ G @ I5 ) )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_o @ I @ I5 )
         => ( ( finite_finite_o @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7809_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_7810_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_7811_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_7812_suminf__le,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_real @ F )
       => ( ( summable_real @ G )
         => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7813_suminf__le,axiom,
    ! [F: nat > nat,G: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_nat @ F )
       => ( ( summable_nat @ G )
         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7814_suminf__le,axiom,
    ! [F: nat > int,G: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( summable_int @ F )
       => ( ( summable_int @ G )
         => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).

% suminf_le
thf(fact_7815_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A4 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7816_sum__cong__Suc,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A4 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A4 )
          = ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7817_ceiling__mono,axiom,
    ! [Y: real,X3: real] :
      ( ( ord_less_eq_real @ Y @ X3 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X3 ) ) ) ).

% ceiling_mono
thf(fact_7818_ceiling__mono,axiom,
    ! [Y: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y @ X3 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).

% ceiling_mono
thf(fact_7819_le__of__int__ceiling,axiom,
    ! [X3: real] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ).

% le_of_int_ceiling
thf(fact_7820_le__of__int__ceiling,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).

% le_of_int_ceiling
thf(fact_7821_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_7822_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_7823_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_7824_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
            @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7825_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
            @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7826_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
            @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) ) ) ) ) ) ).

% suminf_add
thf(fact_7827_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7828_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7829_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M3: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M3 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7830_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E )
       => ~ ! [N8: nat] :
              ~ ! [M3: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M3 )
                 => ! [N9: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_7831_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7832_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M2: nat,K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7833_suminf__nonneg,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7834_suminf__nonneg,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7835_suminf__nonneg,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).

% suminf_nonneg
thf(fact_7836_suminf__eq__zero__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ( suminf_real @ F )
            = zero_zero_real )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_real ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7837_suminf__eq__zero__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ( suminf_nat @ F )
            = zero_zero_nat )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_nat ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7838_suminf__eq__zero__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ( suminf_int @ F )
            = zero_zero_int )
          = ( ! [N3: nat] :
                ( ( F @ N3 )
                = zero_zero_int ) ) ) ) ) ).

% suminf_eq_zero_iff
thf(fact_7839_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 )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7840_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ A4 )
            = zero_zero_real )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7841_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 )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7842_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 )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7843_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 )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7844_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ A4 )
            = zero_zero_rat )
          = ( ! [X4: $o] :
                ( ( member_o @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7845_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 )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7846_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 )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7847_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 )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7848_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 )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7849_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_7850_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_7851_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_7852_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > rat,I: nat > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7853_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > rat,I: complex > complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_rat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7854_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > nat,I: complex > complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ S ) @ ( groups5693394587270226106ex_nat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7855_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > int,I: nat > nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7856_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > int,I: complex > nat,F: nat > int] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ S ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7857_sum__le__included,axiom,
    ! [S: set_complex,T: set_nat,G: nat > int,I: nat > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S ) @ ( groups3539618377306564664at_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7858_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > int,I: complex > complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ T )
             => ( ord_less_eq_int @ zero_zero_int @ ( G @ X5 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X5 )
                    & ( ord_less_eq_int @ ( F @ X5 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ S ) @ ( groups5690904116761175830ex_int @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7859_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 ) ) )
       => ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7860_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 ) ) )
       => ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( ord_less_rat @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7861_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 ) ) )
       => ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( ord_less_rat @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7862_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 ) ) )
       => ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( ord_less_nat @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7863_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 ) ) )
       => ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7864_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 ) ) )
       => ( ? [X: complex] :
              ( ( member_complex @ X @ A4 )
              & ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7865_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_nat @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( ord_less_nat @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7866_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X: nat] :
              ( ( member_nat @ X @ A4 )
              & ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7867_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 ) ) )
       => ( ? [X: int] :
              ( ( member_int @ X @ A4 )
              & ( ord_less_int @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7868_sum__strict__mono__ex1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real,G: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ A4 )
           => ( ord_less_eq_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
       => ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A4 )
              & ( ord_less_real @ ( F @ X ) @ ( G @ X ) ) )
         => ( ord_less_real @ ( groups4567486121110086003t_real @ F @ A4 ) @ ( groups4567486121110086003t_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7869_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X22: real,Y23: real] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7870_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7871_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X22: rat,Y23: rat] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7872_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7873_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H: nat > int,G: nat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X16: int,Y15: int,X22: int,Y23: int] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7874_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_complex,H: complex > int,G: complex > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X16: int,Y15: int,X22: int,Y23: int] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups5690904116761175830ex_int @ H @ S3 ) @ ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7875_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_nat,H: nat > nat,G: nat > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X22: nat,Y23: nat] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups3542108847815614940at_nat @ H @ S3 ) @ ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7876_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_complex,H: complex > complex,G: complex > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X16: complex,Y15: complex,X22: complex,Y23: complex] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X22 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups7754918857620584856omplex @ H @ S3 ) @ ( groups7754918857620584856omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7877_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_nat,H: nat > real,G: nat > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X22: real,Y23: real] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X22 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups6591440286371151544t_real @ H @ S3 ) @ ( groups6591440286371151544t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7878_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_int,H: int > int,G: int > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X16: int,Y15: int,X22: int,Y23: int] :
            ( ( ( R @ X16 @ X22 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X22 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( R @ ( H @ X5 ) @ ( G @ X5 ) ) )
           => ( R @ ( groups4538972089207619220nt_int @ H @ S3 ) @ ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7879_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_7880_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_7881_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > real,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A4 )
             => ( ord_less_real @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7882_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_7883_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_7884_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_7885_sum__strict__mono,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( A4 != bot_bot_set_o )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ A4 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( G @ X5 ) ) )
         => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7886_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_7887_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_7888_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_7889_sum_Oinsert__if,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X3 @ A4 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A4 ) )
            = ( groups8097168146408367636l_real @ G @ A4 ) ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A4 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7890_sum_Oinsert__if,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X3 @ A4 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X3 @ A4 ) )
            = ( groups8691415230153176458o_real @ G @ A4 ) ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X3 @ A4 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8691415230153176458o_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7891_sum_Oinsert__if,axiom,
    ! [A4: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X3 @ A4 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A4 ) )
            = ( groups8778361861064173332t_real @ G @ A4 ) ) )
        & ( ~ ( member_int @ X3 @ A4 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A4 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7892_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X3 @ A4 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( groups5808333547571424918x_real @ G @ A4 ) ) )
        & ( ~ ( member_complex @ X3 @ A4 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7893_sum_Oinsert__if,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X3 @ A4 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( groups1300246762558778688al_rat @ G @ A4 ) ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7894_sum_Oinsert__if,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X3 @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
            = ( groups7872700643590313910_o_rat @ G @ A4 ) ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups7872700643590313910_o_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7895_sum_Oinsert__if,axiom,
    ! [A4: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X3 @ A4 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
            = ( groups3906332499630173760nt_rat @ G @ A4 ) ) )
        & ( ~ ( member_int @ X3 @ A4 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7896_sum_Oinsert__if,axiom,
    ! [A4: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X3 @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( groups2906978787729119204at_rat @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X3 @ A4 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7897_sum_Oinsert__if,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X3 @ A4 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( groups5058264527183730370ex_rat @ G @ A4 ) ) )
        & ( ~ ( member_complex @ X3 @ A4 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7898_sum_Oinsert__if,axiom,
    ! [A4: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X3 @ A4 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( groups1935376822645274424al_nat @ G @ A4 ) ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7899_ceiling__le,axiom,
    ! [X3: real,A: int] :
      ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ A ) ) ).

% ceiling_le
thf(fact_7900_ceiling__le,axiom,
    ! [X3: rat,A: int] :
      ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ A ) ) ).

% ceiling_le
thf(fact_7901_ceiling__le__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
      = ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% ceiling_le_iff
thf(fact_7902_ceiling__le__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
      = ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% ceiling_le_iff
thf(fact_7903_summable__zero__power_H,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7904_summable__zero__power_H,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7905_summable__zero__power_H,axiom,
    ! [F: nat > int] :
      ( summable_int
      @ ^ [N3: nat] : ( times_times_int @ ( F @ N3 ) @ ( power_power_int @ zero_zero_int @ N3 ) ) ) ).

% summable_zero_power'
thf(fact_7906_summable__0__powser,axiom,
    ! [F: nat > complex] :
      ( summable_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) ) ) ).

% summable_0_powser
thf(fact_7907_summable__0__powser,axiom,
    ! [F: nat > real] :
      ( summable_real
      @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) ) ) ).

% summable_0_powser
thf(fact_7908_ceiling__add__le,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7909_ceiling__add__le,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7910_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7911_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% powser_split_head(3)
thf(fact_7912_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7913_summable__powser__split__head,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% summable_powser_split_head
thf(fact_7914_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M2: nat,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N3 @ M2 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
      = ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7915_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M2: nat,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N3 @ M2 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
      = ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_7916_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B4: 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 )
            = B4 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7917_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > real,B4: real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = B4 )
         => ( ( member_o @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7918_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B4: 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 )
            = B4 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7919_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B4: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B4 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7920_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B4: 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 )
            = B4 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7921_sum__nonneg__leq__bound,axiom,
    ! [S: set_o,F: $o > rat,B4: rat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S )
            = B4 )
         => ( ( member_o @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7922_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B4: 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 )
            = B4 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7923_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B4: 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 )
            = B4 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7924_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B4: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B4 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7925_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B4: 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 )
            = B4 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B4 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7926_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_7927_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > real,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8691415230153176458o_real @ F @ S )
            = zero_zero_real )
         => ( ( member_o @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7928_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_7929_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_7930_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_7931_sum__nonneg__0,axiom,
    ! [S: set_o,F: $o > rat,I: $o] :
      ( ( finite_finite_o @ S )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups7872700643590313910_o_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_o @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7932_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_7933_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_7934_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_7935_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_7936_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) ) ).

% summable_norm_comparison_test
thf(fact_7937_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N2 ) ) @ ( G @ N2 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real
          @ ^ [N3: nat] : ( abs_abs_real @ ( F @ N3 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_7938_sum__power__add,axiom,
    ! [X3: complex,M2: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7939_sum__power__add,axiom,
    ! [X3: rat,M2: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7940_sum__power__add,axiom,
    ! [X3: int,M2: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X3 @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7941_sum__power__add,axiom,
    ! [X3: real,M2: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M2 @ I4 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7942_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7943_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N: nat,M2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7944_suminf__pos2,axiom,
    ! [F: nat > real,I: nat] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7945_suminf__pos2,axiom,
    ! [F: nat > nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7946_suminf__pos2,axiom,
    ! [F: nat > int,I: nat] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_7947_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I4: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7948_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I4: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7949_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I4: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I4 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_7950_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_7951_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7952_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_7953_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_7954_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_7955_sum__pos2,axiom,
    ! [I5: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7956_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_7957_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_7958_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_7959_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_7960_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_7961_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_7962_sum__pos,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8691415230153176458o_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7963_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_7964_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_7965_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_7966_sum__pos,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups7872700643590313910_o_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7967_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_7968_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_7969_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_7970_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7971_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G @ T3 )
              = ( groups8691415230153176458o_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7972_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ T3 )
              = ( groups8778361861064173332t_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7973_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7974_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7975_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G @ T3 )
              = ( groups7872700643590313910_o_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7976_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat,H: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
              = ( groups3906332499630173760nt_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7977_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7978_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > nat,H: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7979_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ T3 )
              = ( groups8507830703676809646_o_nat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7980_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7981_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > real,G: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8691415230153176458o_real @ G @ S3 )
              = ( groups8691415230153176458o_real @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7982_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8778361861064173332t_real @ G @ S3 )
              = ( groups8778361861064173332t_real @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7983_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7984_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7985_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups7872700643590313910_o_rat @ G @ S3 )
              = ( groups7872700643590313910_o_rat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7986_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H: int > rat,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
              = ( groups3906332499630173760nt_rat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7987_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_rat ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
              = ( groups5058264527183730370ex_rat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7988_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7989_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = zero_zero_nat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups8507830703676809646_o_nat @ G @ S3 )
              = ( groups8507830703676809646_o_nat @ H @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7990_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7991_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7992_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7993_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7994_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7995_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7996_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ T3 )
            = ( groups3542108847815614940at_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7997_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ T3 )
            = ( groups7754918857620584856omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7998_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups6591440286371151544t_real @ G @ T3 )
            = ( groups6591440286371151544t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7999_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ T3 )
            = ( groups4538972089207619220nt_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_8000_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8001_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8002_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8003_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8004_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S3 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8005_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S3 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8006_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ S3 )
            = ( groups3542108847815614940at_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8007_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ S3 )
            = ( groups7754918857620584856omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8008_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups6591440286371151544t_real @ G @ S3 )
            = ( groups6591440286371151544t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8009_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > int] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ S3 )
            = ( groups4538972089207619220nt_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_8010_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C2 )
                  = ( groups8097168146408367636l_real @ H @ C2 ) )
               => ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8011_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G @ C2 )
                  = ( groups8691415230153176458o_real @ H @ C2 ) )
               => ( ( groups8691415230153176458o_real @ G @ A4 )
                  = ( groups8691415230153176458o_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8012_sum_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ C2 )
                  = ( groups8778361861064173332t_real @ H @ C2 ) )
               => ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8013_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C2 )
                  = ( groups5808333547571424918x_real @ H @ C2 ) )
               => ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8014_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C2 )
                  = ( groups1300246762558778688al_rat @ H @ C2 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8015_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ C2 )
                  = ( groups7872700643590313910_o_rat @ H @ C2 ) )
               => ( ( groups7872700643590313910_o_rat @ G @ A4 )
                  = ( groups7872700643590313910_o_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8016_sum_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > rat,H: int > rat] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ C2 )
                  = ( groups3906332499630173760nt_rat @ H @ C2 ) )
               => ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8017_sum_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C2 )
                  = ( groups5058264527183730370ex_rat @ H @ C2 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8018_sum_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > nat,H: real > nat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C2 )
                  = ( groups1935376822645274424al_nat @ H @ C2 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8019_sum_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ C2 )
                  = ( groups8507830703676809646_o_nat @ H @ C2 ) )
               => ( ( groups8507830703676809646_o_nat @ G @ A4 )
                  = ( groups8507830703676809646_o_nat @ H @ B4 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_8020_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A4 )
                  = ( groups8097168146408367636l_real @ H @ B4 ) )
                = ( ( groups8097168146408367636l_real @ G @ C2 )
                  = ( groups8097168146408367636l_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8021_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8691415230153176458o_real @ G @ A4 )
                  = ( groups8691415230153176458o_real @ H @ B4 ) )
                = ( ( groups8691415230153176458o_real @ G @ C2 )
                  = ( groups8691415230153176458o_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8022_sum_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G @ A4 )
                  = ( groups8778361861064173332t_real @ H @ B4 ) )
                = ( ( groups8778361861064173332t_real @ G @ C2 )
                  = ( groups8778361861064173332t_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8023_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A4 )
                  = ( groups5808333547571424918x_real @ H @ B4 ) )
                = ( ( groups5808333547571424918x_real @ G @ C2 )
                  = ( groups5808333547571424918x_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8024_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A4 )
                  = ( groups1300246762558778688al_rat @ H @ B4 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C2 )
                  = ( groups1300246762558778688al_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8025_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups7872700643590313910_o_rat @ G @ A4 )
                  = ( groups7872700643590313910_o_rat @ H @ B4 ) )
                = ( ( groups7872700643590313910_o_rat @ G @ C2 )
                  = ( groups7872700643590313910_o_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8026_sum_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > rat,H: int > rat] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G @ A4 )
                  = ( groups3906332499630173760nt_rat @ H @ B4 ) )
                = ( ( groups3906332499630173760nt_rat @ G @ C2 )
                  = ( groups3906332499630173760nt_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8027_sum_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A4 )
                  = ( groups5058264527183730370ex_rat @ H @ B4 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C2 )
                  = ( groups5058264527183730370ex_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8028_sum_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > nat,H: real > nat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A4 )
                  = ( groups1935376822645274424al_nat @ H @ B4 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C2 )
                  = ( groups1935376822645274424al_nat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8029_sum_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > nat,H: $o > nat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups8507830703676809646_o_nat @ G @ A4 )
                  = ( groups8507830703676809646_o_nat @ H @ B4 ) )
                = ( ( groups8507830703676809646_o_nat @ G @ C2 )
                  = ( groups8507830703676809646_o_nat @ H @ C2 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_8030_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8031_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8032_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5693394587270226106ex_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8033_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups5690904116761175830ex_int @ G @ A4 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8034_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups2906978787729119204at_rat @ G @ A4 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8035_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3539618377306564664at_int @ G @ A4 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8036_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups3542108847815614940at_nat @ G @ A4 )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8037_sum_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > complex] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups7754918857620584856omplex @ G @ A4 )
          = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8038_sum_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups6591440286371151544t_real @ G @ A4 )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8039_sum_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups4538972089207619220nt_int @ G @ A4 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ B4 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_8040_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8041_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8042_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8043_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8044_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8045_sum__diff,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8046_sum__diff,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8047_sum__diff,axiom,
    ! [A4: set_int,B4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_set_int @ B4 @ A4 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B4 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8048_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
       => ( ( groups4567486121110086003t_real @ F @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
          = ( minus_minus_real @ ( groups4567486121110086003t_real @ F @ A4 ) @ ( groups4567486121110086003t_real @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8049_sum__diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
       => ( ( groups342789780944988191at_rat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( groups342789780944988191at_rat @ F @ A4 ) @ ( groups342789780944988191at_rat @ F @ B4 ) ) ) ) ) ).

% sum_diff
thf(fact_8050_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A4 ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8051_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8052_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8053_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A4 ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8054_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8055_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8056_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8057_sum_Ounion__inter,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( sup_sup_set_complex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) )
          = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8058_sum_Ounion__inter,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( sup_sup_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8059_sum_Ounion__inter,axiom,
    ! [A4: set_int,B4: set_int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B4 ) ) )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B4 ) ) ) ) ) ).

% sum.union_inter
thf(fact_8060_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: complex > real,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G @ A4 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8061_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: complex > rat,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8062_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: complex > nat,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G @ A4 )
        = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8063_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: complex > int,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5690904116761175830ex_int @ G @ A4 )
        = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8064_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > rat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G @ A4 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8065_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > int,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int @ G @ A4 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8066_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > nat,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3542108847815614940at_nat @ G @ A4 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8067_sum_OInt__Diff,axiom,
    ! [A4: set_complex,G: complex > complex,B4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex @ G @ A4 )
        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8068_sum_OInt__Diff,axiom,
    ! [A4: set_nat,G: nat > real,B4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6591440286371151544t_real @ G @ A4 )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8069_sum_OInt__Diff,axiom,
    ! [A4: set_int,G: int > int,B4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4538972089207619220nt_int @ G @ A4 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) ) ) ) ).

% sum.Int_Diff
thf(fact_8070_of__int__ceiling__le__add__one,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_8071_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_8072_of__int__ceiling__diff__one__le,axiom,
    ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_8073_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_8074_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8075_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > int,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8076_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > nat,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8077_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_8078_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_8079_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_8080_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_8081_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_8082_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_rat @ ( G @ M2 ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8083_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_int @ ( G @ M2 ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8084_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_nat @ ( G @ M2 ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8085_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_real @ ( G @ M2 ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_8086_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8087_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8088_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8089_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_8090_powser__inside,axiom,
    ! [F: nat > real,X3: real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_8091_powser__inside,axiom,
    ! [F: nat > complex,X3: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) ) ) ) ).

% powser_inside
thf(fact_8092_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_8093_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_8094_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_8095_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_8096_suminf__split__head,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).

% suminf_split_head
thf(fact_8097_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: complex > $o,H: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( if_real @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_real @ ( groups5808333547571424918x_real @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8098_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: complex > $o,H: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X4: complex] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8099_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: complex > $o,H: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8100_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: complex > $o,H: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5690904116761175830ex_int
          @ ^ [X4: complex] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups5690904116761175830ex_int @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) ) @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8101_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: nat > $o,H: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8102_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: nat > $o,H: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3539618377306564664at_int
          @ ^ [X4: nat] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8103_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: nat > $o,H: nat > nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( if_nat @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8104_sum_OIf__cases,axiom,
    ! [A4: set_complex,P: complex > $o,H: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : ( if_complex @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P ) ) ) @ ( groups7754918857620584856omplex @ G @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8105_sum_OIf__cases,axiom,
    ! [A4: set_nat,P: nat > $o,H: nat > real,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups6591440286371151544t_real
          @ ^ [X4: nat] : ( if_real @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P ) ) ) @ ( groups6591440286371151544t_real @ G @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8106_sum_OIf__cases,axiom,
    ! [A4: set_int,P: int > $o,H: int > int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : ( if_int @ ( P @ X4 ) @ ( H @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( plus_plus_int @ ( groups4538972089207619220nt_int @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ) ) ) ) ).

% sum.If_cases
thf(fact_8107_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G @ M2 )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8108_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G @ M2 )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8109_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G @ M2 )
          @ ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8110_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G @ M2 )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_8111_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_8112_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_8113_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( minus_minus_real @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_8114_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8115_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > real] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ ( groups8691415230153176458o_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8116_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8117_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8118_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8119_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8120_sum__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8121_sum__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8122_sum__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8123_sum__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > nat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ F @ B4 ) ) ) ) ) ).

% sum_mono2
thf(fact_8124_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A4 ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8125_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8126_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8127_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A4 ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8128_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8129_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8130_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_nat ) )
         => ( ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A4 ) @ ( groups3542108847815614940at_nat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8131_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( inf_inf_set_complex @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A4 ) @ ( groups7754918857620584856omplex @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8132_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( inf_inf_set_nat @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_real ) )
         => ( ( groups6591440286371151544t_real @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A4 ) @ ( groups6591440286371151544t_real @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8133_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B4: set_int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( inf_inf_set_int @ A4 @ B4 ) )
             => ( ( G @ X5 )
                = zero_zero_int ) )
         => ( ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A4 ) @ ( groups4538972089207619220nt_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_8134_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > real,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8135_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > rat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8136_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > nat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8137_sum_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > int,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8138_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > real,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8691415230153176458o_real @ G @ ( insert_o @ X3 @ A4 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8139_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > rat,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups7872700643590313910_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8140_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > nat,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8507830703676809646_o_nat @ G @ ( insert_o @ X3 @ A4 ) )
        = ( plus_plus_nat @ ( G @ X3 ) @ ( groups8507830703676809646_o_nat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8141_sum_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > int,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups8505340233167759370_o_int @ G @ ( insert_o @ X3 @ A4 ) )
        = ( plus_plus_int @ ( G @ X3 ) @ ( groups8505340233167759370_o_int @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8142_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > real,X3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A4 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8143_sum_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > rat,X3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_8144_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups8097168146408367636l_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8145_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups5808333547571424918x_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8146_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups1300246762558778688al_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8147_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups5058264527183730370ex_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8148_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups1935376822645274424al_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8149_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups5693394587270226106ex_nat @ G @ A4 )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8150_sum_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > int] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups1932886352136224148al_int @ G @ A4 )
          = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8151_sum_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups5690904116761175830ex_int @ G @ A4 )
          = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8152_sum_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X3 @ A4 )
       => ( ( groups8691415230153176458o_real @ G @ A4 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8691415230153176458o_real @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8153_sum_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X3 @ A4 )
       => ( ( groups7872700643590313910_o_rat @ G @ A4 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups7872700643590313910_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_8154_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ A @ A4 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A4 )
         => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8155_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ A @ A4 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A4 )
         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8156_sum__diff1,axiom,
    ! [A4: set_o,A: $o,F: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ A @ A4 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o @ A @ A4 )
         => ( ( groups8691415230153176458o_real @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( groups8691415230153176458o_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8157_sum__diff1,axiom,
    ! [A4: set_int,A: int,F: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ A @ A4 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A4 )
         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8158_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ A @ A4 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A4 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8159_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ A @ A4 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A4 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8160_sum__diff1,axiom,
    ! [A4: set_o,A: $o,F: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ A @ A4 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( minus_minus_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_o @ A @ A4 )
         => ( ( groups7872700643590313910_o_rat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
            = ( groups7872700643590313910_o_rat @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8161_sum__diff1,axiom,
    ! [A4: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ A @ A4 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A4 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8162_sum__diff1,axiom,
    ! [A4: set_real,A: real,F: real > int] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ A @ A4 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A4 )
         => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1932886352136224148al_int @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8163_sum__diff1,axiom,
    ! [A4: set_complex,A: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ A @ A4 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A4 )
         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5690904116761175830ex_int @ F @ A4 ) ) ) ) ) ).

% sum_diff1
thf(fact_8164_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8165_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8166_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ A4 ) @ ( groups2906978787729119204at_rat @ F @ B4 ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8167_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ ( groups5690904116761175830ex_int @ F @ B4 ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8168_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ ( groups3539618377306564664at_int @ F @ B4 ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8169_sum__Un,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups7754918857620584856omplex @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( minus_minus_complex @ ( plus_plus_complex @ ( groups7754918857620584856omplex @ F @ A4 ) @ ( groups7754918857620584856omplex @ F @ B4 ) ) @ ( groups7754918857620584856omplex @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8170_sum__Un,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups6591440286371151544t_real @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ A4 ) @ ( groups6591440286371151544t_real @ F @ B4 ) ) @ ( groups6591440286371151544t_real @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8171_sum__Un,axiom,
    ! [A4: set_int,B4: set_int,F: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A4 @ B4 ) )
          = ( minus_minus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ ( groups4538972089207619220nt_int @ F @ B4 ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8172_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > real] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ( groups4567486121110086003t_real @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( groups4567486121110086003t_real @ F @ A4 ) @ ( groups4567486121110086003t_real @ F @ B4 ) ) @ ( groups4567486121110086003t_real @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8173_sum__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > rat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ( groups342789780944988191at_rat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( groups342789780944988191at_rat @ F @ A4 ) @ ( groups342789780944988191at_rat @ F @ B4 ) ) @ ( groups342789780944988191at_rat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un
thf(fact_8174_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ A4 ) @ ( groups5808333547571424918x_real @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8175_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ A4 ) @ ( groups5058264527183730370ex_rat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8176_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8177_sum_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ A4 ) @ ( groups5690904116761175830ex_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8178_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups8691415230153176458o_real @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( plus_plus_real @ ( groups8691415230153176458o_real @ G @ A4 ) @ ( groups8691415230153176458o_real @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8179_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups7872700643590313910_o_rat @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( plus_plus_rat @ ( groups7872700643590313910_o_rat @ G @ A4 ) @ ( groups7872700643590313910_o_rat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8180_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups8507830703676809646_o_nat @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( plus_plus_nat @ ( groups8507830703676809646_o_nat @ G @ A4 ) @ ( groups8507830703676809646_o_nat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8181_sum_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > int] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups8505340233167759370_o_int @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups8505340233167759370_o_int @ G @ A4 ) @ ( groups8505340233167759370_o_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8182_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( inf_inf_set_nat @ A4 @ B4 )
            = bot_bot_set_nat )
         => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ A4 ) @ ( groups2906978787729119204at_rat @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8183_sum_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( inf_inf_set_nat @ A4 @ B4 )
            = bot_bot_set_nat )
         => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ A4 ) @ ( groups3539618377306564664at_int @ G @ B4 ) ) ) ) ) ) ).

% sum.union_disjoint
thf(fact_8184_ceiling__correct,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) @ one_one_real ) @ X3 )
      & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).

% ceiling_correct
thf(fact_8185_ceiling__correct,axiom,
    ! [X3: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) @ one_one_rat ) @ X3 )
      & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ) ).

% ceiling_correct
thf(fact_8186_ceiling__unique,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) )
       => ( ( archim7802044766580827645g_real @ X3 )
          = Z2 ) ) ) ).

% ceiling_unique
thf(fact_8187_ceiling__unique,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) )
       => ( ( archim2889992004027027881ng_rat @ X3 )
          = Z2 ) ) ) ).

% ceiling_unique
thf(fact_8188_ceiling__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X3 )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X3 )
        & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_8189_ceiling__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X3 )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X3 )
        & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_8190_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_8191_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I4 ) ) )
           => ( P @ I4 ) ) ) ) ).

% ceiling_split
thf(fact_8192_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ( groups5808333547571424918x_real @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5808333547571424918x_real @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8193_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ( groups5058264527183730370ex_rat @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5058264527183730370ex_rat @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8194_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8195_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ( groups5690904116761175830ex_int @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5690904116761175830ex_int @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8196_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ( groups2906978787729119204at_rat @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
        = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups2906978787729119204at_rat @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8197_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ( groups3539618377306564664at_int @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups3539618377306564664at_int @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8198_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
        = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8199_sum__Un2,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ ( sup_sup_set_complex @ A4 @ B4 ) )
     => ( ( groups7754918857620584856omplex @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
        = ( plus_plus_complex @ ( plus_plus_complex @ ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups7754918857620584856omplex @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8200_sum__Un2,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ( groups6591440286371151544t_real @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
        = ( plus_plus_real @ ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups6591440286371151544t_real @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8201_sum__Un2,axiom,
    ! [A4: set_int,B4: set_int,F: int > int] :
      ( ( finite_finite_int @ ( sup_sup_set_int @ A4 @ B4 ) )
     => ( ( groups4538972089207619220nt_int @ F @ ( sup_sup_set_int @ A4 @ B4 ) )
        = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ B4 @ A4 ) ) ) @ ( groups4538972089207619220nt_int @ F @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ) ).

% sum_Un2
thf(fact_8202_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5808333547571424918x_real @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5808333547571424918x_real @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8203_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5058264527183730370ex_rat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8204_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5693394587270226106ex_nat @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8205_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5690904116761175830ex_int @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups5690904116761175830ex_int @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8206_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups2906978787729119204at_rat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( plus_plus_rat @ ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups2906978787729119204at_rat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8207_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups3539618377306564664at_int @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups3539618377306564664at_int @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8208_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups3542108847815614940at_nat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( plus_plus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups3542108847815614940at_nat @ G @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups3542108847815614940at_nat @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8209_sum_Ounion__diff2,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups7754918857620584856omplex @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( plus_plus_complex @ ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups7754918857620584856omplex @ G @ ( minus_811609699411566653omplex @ B4 @ A4 ) ) ) @ ( groups7754918857620584856omplex @ G @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8210_sum_Ounion__diff2,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups6591440286371151544t_real @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( plus_plus_real @ ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups6591440286371151544t_real @ G @ ( minus_minus_set_nat @ B4 @ A4 ) ) ) @ ( groups6591440286371151544t_real @ G @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8211_sum_Ounion__diff2,axiom,
    ! [A4: set_int,B4: set_int,G: int > int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( groups4538972089207619220nt_int @ G @ ( sup_sup_set_int @ A4 @ B4 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ B4 @ A4 ) ) ) @ ( groups4538972089207619220nt_int @ G @ ( inf_inf_set_int @ A4 @ B4 ) ) ) ) ) ) ).

% sum.union_diff2
thf(fact_8212_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_8213_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_8214_ceiling__less__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_8215_ceiling__less__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_8216_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8217_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8218_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8219_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_8220_sum__div__partition,axiom,
    ! [A4: set_complex,F: complex > code_integer,B: code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( divide6298287555418463151nteger @ ( groups6621422865394947399nteger @ F @ A4 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups6621422865394947399nteger
            @ ^ [A6: complex] : ( divide6298287555418463151nteger @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_complex @ A4
              @ ( collect_complex
                @ ^ [A6: complex] : ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups6621422865394947399nteger @ F
              @ ( inf_inf_set_complex @ A4
                @ ( collect_complex
                  @ ^ [A6: complex] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8221_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > code_integer,B: code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide6298287555418463151nteger @ ( groups7501900531339628137nteger @ F @ A4 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups7501900531339628137nteger
            @ ^ [A6: nat] : ( divide6298287555418463151nteger @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A6: nat] : ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups7501900531339628137nteger @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8222_sum__div__partition,axiom,
    ! [A4: set_complex,F: complex > nat,B: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( divide_divide_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups5693394587270226106ex_nat
            @ ^ [A6: complex] : ( divide_divide_nat @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_complex @ A4
              @ ( collect_complex
                @ ^ [A6: complex] : ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups5693394587270226106ex_nat @ F
              @ ( inf_inf_set_complex @ A4
                @ ( collect_complex
                  @ ^ [A6: complex] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8223_sum__div__partition,axiom,
    ! [A4: set_complex,F: complex > int,B: int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( divide_divide_int @ ( groups5690904116761175830ex_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups5690904116761175830ex_int
            @ ^ [A6: complex] : ( divide_divide_int @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_complex @ A4
              @ ( collect_complex
                @ ^ [A6: complex] : ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups5690904116761175830ex_int @ F
              @ ( inf_inf_set_complex @ A4
                @ ( collect_complex
                  @ ^ [A6: complex] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8224_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > int,B: int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide_divide_int @ ( groups3539618377306564664at_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups3539618377306564664at_int
            @ ^ [A6: nat] : ( divide_divide_int @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A6: nat] : ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups3539618377306564664at_int @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8225_sum__div__partition,axiom,
    ! [A4: set_nat,F: nat > nat,B: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( divide_divide_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups3542108847815614940at_nat
            @ ^ [A6: nat] : ( divide_divide_nat @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_nat @ A4
              @ ( collect_nat
                @ ^ [A6: nat] : ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups3542108847815614940at_nat @ F
              @ ( inf_inf_set_nat @ A4
                @ ( collect_nat
                  @ ^ [A6: nat] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8226_sum__div__partition,axiom,
    ! [A4: set_int,F: int > int,B: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( divide_divide_int @ ( groups4538972089207619220nt_int @ F @ A4 ) @ B )
        = ( plus_plus_int
          @ ( groups4538972089207619220nt_int
            @ ^ [A6: int] : ( divide_divide_int @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_int @ A4
              @ ( collect_int
                @ ^ [A6: int] : ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_int
            @ ( groups4538972089207619220nt_int @ F
              @ ( inf_inf_set_int @ A4
                @ ( collect_int
                  @ ^ [A6: int] :
                      ~ ( dvd_dvd_int @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8227_sum__div__partition,axiom,
    ! [A4: set_set_nat,F: set_nat > code_integer,B: code_integer] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( divide6298287555418463151nteger @ ( groups9190459664516455967nteger @ F @ A4 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups9190459664516455967nteger
            @ ^ [A6: set_nat] : ( divide6298287555418463151nteger @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_set_nat @ A4
              @ ( collect_set_nat
                @ ^ [A6: set_nat] : ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups9190459664516455967nteger @ F
              @ ( inf_inf_set_set_nat @ A4
                @ ( collect_set_nat
                  @ ^ [A6: set_nat] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8228_sum__div__partition,axiom,
    ! [A4: set_list_nat,F: list_nat > code_integer,B: code_integer] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( divide6298287555418463151nteger @ ( groups3257113351181983737nteger @ F @ A4 ) @ B )
        = ( plus_p5714425477246183910nteger
          @ ( groups3257113351181983737nteger
            @ ^ [A6: list_nat] : ( divide6298287555418463151nteger @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_list_nat @ A4
              @ ( collect_list_nat
                @ ^ [A6: list_nat] : ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide6298287555418463151nteger
            @ ( groups3257113351181983737nteger @ F
              @ ( inf_inf_set_list_nat @ A4
                @ ( collect_list_nat
                  @ ^ [A6: list_nat] :
                      ~ ( dvd_dvd_Code_integer @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8229_sum__div__partition,axiom,
    ! [A4: set_set_nat,F: set_nat > nat,B: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( divide_divide_nat @ ( groups8294997508430121362at_nat @ F @ A4 ) @ B )
        = ( plus_plus_nat
          @ ( groups8294997508430121362at_nat
            @ ^ [A6: set_nat] : ( divide_divide_nat @ ( F @ A6 ) @ B )
            @ ( inf_inf_set_set_nat @ A4
              @ ( collect_set_nat
                @ ^ [A6: set_nat] : ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
          @ ( divide_divide_nat
            @ ( groups8294997508430121362at_nat @ F
              @ ( inf_inf_set_set_nat @ A4
                @ ( collect_set_nat
                  @ ^ [A6: set_nat] :
                      ~ ( dvd_dvd_nat @ B @ ( F @ A6 ) ) ) ) )
            @ B ) ) ) ) ).

% sum_div_partition
thf(fact_8230_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8231_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8232_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8233_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8234_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8235_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8236_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > int,C: real > int] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8237_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8238_sum_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > real,C: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups8691415230153176458o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8691415230153176458o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8239_sum_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > rat,C: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups7872700643590313910_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups7872700643590313910_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_8240_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_8241_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A4 ) @ ( groups8097168146408367636l_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8242_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: $o > real] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8691415230153176458o_real @ F @ A4 ) @ ( groups8691415230153176458o_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8243_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A4 ) @ ( groups8778361861064173332t_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8244_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B4 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A4 ) @ ( groups5808333547571424918x_real @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8245_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A4 ) @ ( groups1300246762558778688al_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8246_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: $o > rat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups7872700643590313910_o_rat @ F @ A4 ) @ ( groups7872700643590313910_o_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8247_sum__strict__mono2,axiom,
    ! [B4: set_int,A4: set_int,B: int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: int] :
                  ( ( member_int @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A4 ) @ ( groups3906332499630173760nt_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8248_sum__strict__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B4 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X5: complex] :
                  ( ( member_complex @ X5 @ B4 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A4 ) @ ( groups5058264527183730370ex_rat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8249_sum__strict__mono2,axiom,
    ! [B4: set_real,A4: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: real] :
                  ( ( member_real @ X5 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8250_sum__strict__mono2,axiom,
    ! [B4: set_o,A4: set_o,B: $o,F: $o > nat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ( member_o @ B @ ( minus_minus_set_o @ B4 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X5: $o] :
                  ( ( member_o @ X5 @ B4 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
             => ( ord_less_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ F @ B4 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_8251_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > real] :
      ( ( member_real @ I @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8252_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8253_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: $o > real] :
      ( ( member_o @ I @ A4 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8691415230153176458o_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8254_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: int > real] :
      ( ( member_int @ I @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8255_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > rat] :
      ( ( member_real @ I @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8256_member__le__sum,axiom,
    ! [I: complex,A4: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A4 )
     => ( ! [X5: complex] :
            ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8257_member__le__sum,axiom,
    ! [I: $o,A4: set_o,F: $o > rat] :
      ( ( member_o @ I @ A4 )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( minus_minus_set_o @ A4 @ ( insert_o @ I @ bot_bot_set_o ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_o @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups7872700643590313910_o_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8258_member__le__sum,axiom,
    ! [I: int,A4: set_int,F: int > rat] :
      ( ( member_int @ I @ A4 )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8259_member__le__sum,axiom,
    ! [I: nat,A4: set_nat,F: nat > rat] :
      ( ( member_nat @ I @ A4 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
       => ( ( finite_finite_nat @ A4 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8260_member__le__sum,axiom,
    ! [I: real,A4: set_real,F: real > nat] :
      ( ( member_real @ I @ A4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_8261_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_8262_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_8263_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_8264_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_8265_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) ) )
          @ Z2 )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N3: nat] : ( times_times_complex @ ( F @ N3 ) @ ( power_power_complex @ Z2 @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8266_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) ) )
          @ Z2 )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ Z2 @ N3 ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8267_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( 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 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8268_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( 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 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8269_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: complex > code_integer,A: complex > code_integer,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I4: complex] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8270_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I4: real] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8271_convex__sum__bound__le,axiom,
    ! [I5: set_o,X3: $o > code_integer,A: $o > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups4406642042086082107nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups4406642042086082107nteger
                  @ ^ [I4: $o] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8272_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I4: nat] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8273_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I4: int] : ( times_3573771949741848930nteger @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8274_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: complex > real,A: complex > real,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I4: complex] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8275_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > real,A: real > real,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I4: real] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8276_convex__sum__bound__le,axiom,
    ! [I5: set_o,X3: $o > real,A: $o > real,B: real,Delta: real] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups8691415230153176458o_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8691415230153176458o_real
                  @ ^ [I4: $o] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8277_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > real,A: int > real,B: 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 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I4: int] : ( times_times_real @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8278_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X3: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I4: complex] : ( times_times_rat @ ( A @ I4 ) @ ( X3 @ I4 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8279_summable__power__series,axiom,
    ! [F: nat > real,Z2: 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 @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( summable_real
              @ ^ [I4: nat] : ( times_times_real @ ( F @ I4 ) @ ( power_power_real @ Z2 @ I4 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8280_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N2 ) ) @ ( power_power_real @ R0 @ N2 ) ) @ M7 )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R2 @ N3 ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_8281_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_8282_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_int @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_8283_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_real @ ( F @ M2 ) @ ( F @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_8284_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_rat @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_8285_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_int @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_8286_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_real @ ( F @ N ) @ ( F @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_8287_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N2 ) ) ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_ratio_test
thf(fact_8288_summable__ratio__test,axiom,
    ! [C: real,N5: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N2: nat] :
            ( ( ord_less_eq_nat @ N5 @ N2 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N2 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N2 ) ) ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_ratio_test
thf(fact_8289_ceiling__eq,axiom,
    ! [N: int,X3: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X3 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_8290_ceiling__eq,axiom,
    ! [N: int,X3: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X3 )
          = ( plus_plus_int @ N @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_8291_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_8292_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_8293_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X3: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X3 @ M2 ) @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8294_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X3: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X3 @ M2 ) @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8295_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X3: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X3 @ M2 ) @ ( power_power_int @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8296_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X3: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X3 @ M2 ) @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_8297_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_8298_sum_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_8299_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_8300_sum_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_8301_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_8302_gauss__sum__nat,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( 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_8303_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_8304_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_8305_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_8306_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_8307_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_8308_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_8309_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_8310_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_8311_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_8312_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_8313_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_8314_Sum__Icc__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( plus_plus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_8315_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_8316_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_8317_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_8318_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_8319_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_8320_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_8321_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_8322_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_8323_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_8324_sum__gp__offset,axiom,
    ! [X3: complex,M2: nat,N: nat] :
      ( ( ( X3 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X3 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8325_sum__gp__offset,axiom,
    ! [X3: rat,M2: nat,N: nat] :
      ( ( ( X3 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X3 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8326_sum__gp__offset,axiom,
    ! [X3: real,M2: nat,N: nat] :
      ( ( ( X3 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X3 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_8327_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_8328_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_8329_lemma__termdiff2,axiom,
    ! [H: complex,Z2: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ N ) @ ( power_power_complex @ Z2 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q4: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ Q4 ) @ ( power_power_complex @ Z2 @ ( 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_8330_lemma__termdiff2,axiom,
    ! [H: rat,Z2: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H ) @ N ) @ ( power_power_rat @ Z2 @ N ) ) @ H ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q4: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H ) @ Q4 ) @ ( power_power_rat @ Z2 @ ( 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_8331_lemma__termdiff2,axiom,
    ! [H: real,Z2: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ N ) @ ( power_power_real @ Z2 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q4: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ Q4 ) @ ( power_power_real @ Z2 @ ( 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_8332_ceiling__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X3 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K2 ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8333_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( power_power_real @ Z2 @ N3 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8334_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N3 ) ) @ ( power_power_complex @ Z2 @ N3 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8335_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8336_monoseq__def,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X8: nat > set_nat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_set_nat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_set_nat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8337_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X8: nat > rat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8338_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X8: nat > num] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_num @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8339_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_nat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8340_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_int @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
          | ! [M5: nat,N3: nat] :
              ( ( ord_less_eq_nat @ M5 @ N3 )
             => ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8341_monoI2,axiom,
    ! [X6: nat > real] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI2
thf(fact_8342_monoI2,axiom,
    ! [X6: nat > set_nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_set_nat @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% monoI2
thf(fact_8343_monoI2,axiom,
    ! [X6: nat > rat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI2
thf(fact_8344_monoI2,axiom,
    ! [X6: nat > num] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI2
thf(fact_8345_monoI2,axiom,
    ! [X6: nat > nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI2
thf(fact_8346_monoI2,axiom,
    ! [X6: nat > int] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ M ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI2
thf(fact_8347_lessThan__subset__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X3 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8348_lessThan__subset__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X3 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8349_lessThan__subset__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X3 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8350_lessThan__subset__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X3 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8351_lessThan__subset__iff,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X3 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_8352_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_8353_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_8354_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_8355_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_8356_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_8357_single__Diff__lessThan,axiom,
    ! [K2: $o] :
      ( ( minus_minus_set_o @ ( insert_o @ K2 @ bot_bot_set_o ) @ ( set_ord_lessThan_o @ K2 ) )
      = ( insert_o @ K2 @ bot_bot_set_o ) ) ).

% single_Diff_lessThan
thf(fact_8358_single__Diff__lessThan,axiom,
    ! [K2: int] :
      ( ( minus_minus_set_int @ ( insert_int @ K2 @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K2 ) )
      = ( insert_int @ K2 @ bot_bot_set_int ) ) ).

% single_Diff_lessThan
thf(fact_8359_single__Diff__lessThan,axiom,
    ! [K2: nat] :
      ( ( minus_minus_set_nat @ ( insert_nat @ K2 @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K2 ) )
      = ( insert_nat @ K2 @ bot_bot_set_nat ) ) ).

% single_Diff_lessThan
thf(fact_8360_single__Diff__lessThan,axiom,
    ! [K2: real] :
      ( ( minus_minus_set_real @ ( insert_real @ K2 @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K2 ) )
      = ( insert_real @ K2 @ bot_bot_set_real ) ) ).

% single_Diff_lessThan
thf(fact_8361_numeral__powr__numeral__real,axiom,
    ! [M2: num,N: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_real @ N ) )
      = ( power_power_real @ ( numeral_numeral_real @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ).

% numeral_powr_numeral_real
thf(fact_8362_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X3: complex] :
      ( ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_8363_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X3: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_8364_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_8365_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_8366_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T )
         => ( ord_less_eq_real @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8367_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T )
         => ( ord_less_eq_nat @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8368_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( G @ N2 ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T )
         => ( ord_less_eq_int @ S @ T ) ) ) ) ).

% sums_le
thf(fact_8369_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
          @ ^ [X4: real] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_or5984915006950818249n_real @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8370_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
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_8371_lessThan__non__empty,axiom,
    ! [X3: int] :
      ( ( set_ord_lessThan_int @ X3 )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_8372_lessThan__non__empty,axiom,
    ! [X3: real] :
      ( ( set_or5984915006950818249n_real @ X3 )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_8373_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N3: nat] : ( plus_plus_real @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8374_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N3: nat] : ( plus_plus_nat @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8375_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N3: nat] : ( plus_plus_int @ ( F @ N3 ) @ ( G @ N3 ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8376_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_8377_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_8378_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_8379_Iio__eq__empty__iff,axiom,
    ! [N: $o] :
      ( ( ( set_ord_lessThan_o @ N )
        = bot_bot_set_o )
      = ( N = bot_bot_o ) ) ).

% Iio_eq_empty_iff
thf(fact_8380_Iio__eq__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = bot_bot_nat ) ) ).

% Iio_eq_empty_iff
thf(fact_8381_lessThan__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% lessThan_Suc
thf(fact_8382_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_8383_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
          @ ^ [X4: complex] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8384_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
          @ ^ [X4: real] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( groups1935376822645274424al_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8385_sum__subtractf__nat,axiom,
    ! [A4: set_o,G: $o > nat,F: $o > nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups8507830703676809646_o_nat
          @ ^ [X4: $o] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( groups8507830703676809646_o_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8386_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
          @ ^ [X4: int] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( groups4541462559716669496nt_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8387_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
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F @ X4 ) @ ( G @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ G @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_8388_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: complex] :
                  ( ( member_complex @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8389_sum__eq__Suc0__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( ( groups977919841031483927at_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8390_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ A4 )
                 => ( ( X4 != Y3 )
                   => ( ( F @ Y3 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_8391_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_8392_sums__Suc__imp,axiom,
    ! [F: nat > real,S: real] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
          @ S )
       => ( sums_real @ F @ S ) ) ) ).

% sums_Suc_imp
thf(fact_8393_sums__Suc__iff,axiom,
    ! [F: nat > real,S: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_8394_sums__Suc,axiom,
    ! [F: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8395_sums__Suc,axiom,
    ! [F: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8396_sums__Suc,axiom,
    ! [F: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N3: nat] : ( F @ ( suc @ N3 ) )
        @ L )
     => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_8397_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_8398_ivl__disj__int__one_I4_J,axiom,
    ! [L: $o,U: $o] :
      ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ L ) @ ( set_or8904488021354931149Most_o @ L @ U ) )
      = bot_bot_set_o ) ).

% ivl_disj_int_one(4)
thf(fact_8399_ivl__disj__int__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = bot_bot_set_nat ) ).

% ivl_disj_int_one(4)
thf(fact_8400_ivl__disj__int__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
      = bot_bot_set_int ) ).

% ivl_disj_int_one(4)
thf(fact_8401_ivl__disj__int__one_I4_J,axiom,
    ! [L: real,U: real] :
      ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ L ) @ ( set_or1222579329274155063t_real @ L @ U ) )
      = bot_bot_set_real ) ).

% ivl_disj_int_one(4)
thf(fact_8402_powr__add,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( powr_real @ X3 @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ).

% powr_add
thf(fact_8403_lessThan__nat__numeral,axiom,
    ! [K2: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_8404_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_8405_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_8406_sum__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_8407_sum__nth__roots,axiom,
    ! [N: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( groups7754918857620584856omplex
          @ ^ [X4: complex] : X4
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_8408_powser__sums__if,axiom,
    ! [M2: nat,Z2: complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( if_complex @ ( N3 = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N3 ) )
      @ ( power_power_complex @ Z2 @ M2 ) ) ).

% powser_sums_if
thf(fact_8409_powser__sums__if,axiom,
    ! [M2: nat,Z2: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( if_real @ ( N3 = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N3 ) )
      @ ( power_power_real @ Z2 @ M2 ) ) ).

% powser_sums_if
thf(fact_8410_powser__sums__if,axiom,
    ! [M2: nat,Z2: int] :
      ( sums_int
      @ ^ [N3: nat] : ( times_times_int @ ( if_int @ ( N3 = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N3 ) )
      @ ( power_power_int @ Z2 @ M2 ) ) ).

% powser_sums_if
thf(fact_8411_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N3: nat] : ( times_times_complex @ ( A @ N3 ) @ ( power_power_complex @ zero_zero_complex @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8412_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( A @ N3 ) @ ( power_power_real @ zero_zero_real @ N3 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_8413_sum__diff__nat,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ B4 @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8414_sum__diff__nat,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8415_sum__diff__nat,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ B4 @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_8416_sum__diff1__nat,axiom,
    ! [A: produc3843707927480180839at_nat,A4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
          = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ A4 )
       => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A4 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
          = ( groups3860910324918113789at_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8417_sum__diff1__nat,axiom,
    ! [A: complex,A4: set_complex,F: complex > nat] :
      ( ( ( member_complex @ A @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_complex @ A @ A4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
          = ( groups5693394587270226106ex_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8418_sum__diff1__nat,axiom,
    ! [A: real,A4: set_real,F: real > nat] :
      ( ( ( member_real @ A @ A4 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_real @ A @ A4 )
       => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
          = ( groups1935376822645274424al_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8419_sum__diff1__nat,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A4 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
          = ( groups977919841031483927at_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8420_sum__diff1__nat,axiom,
    ! [A: $o,A4: set_o,F: $o > nat] :
      ( ( ( member_o @ A @ A4 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( minus_minus_nat @ ( groups8507830703676809646_o_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_o @ A @ A4 )
       => ( ( groups8507830703676809646_o_nat @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
          = ( groups8507830703676809646_o_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8421_sum__diff1__nat,axiom,
    ! [A: int,A4: set_int,F: int > nat] :
      ( ( ( member_int @ A @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_int @ A @ A4 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
          = ( groups4541462559716669496nt_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8422_sum__diff1__nat,axiom,
    ! [A: nat,A4: set_nat,F: nat > nat] :
      ( ( ( member_nat @ A @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( F @ A ) ) ) )
      & ( ~ ( member_nat @ A @ A4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
          = ( groups3542108847815614940at_nat @ F @ A4 ) ) ) ) ).

% sum_diff1_nat
thf(fact_8423_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_8424_Iio__Int__singleton,axiom,
    ! [X3: $o,K2: $o] :
      ( ( ( ord_less_o @ X3 @ K2 )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K2 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
          = ( insert_o @ X3 @ bot_bot_set_o ) ) )
      & ( ~ ( ord_less_o @ X3 @ K2 )
       => ( ( inf_inf_set_o @ ( set_ord_lessThan_o @ K2 ) @ ( insert_o @ X3 @ bot_bot_set_o ) )
          = bot_bot_set_o ) ) ) ).

% Iio_Int_singleton
thf(fact_8425_Iio__Int__singleton,axiom,
    ! [X3: rat,K2: rat] :
      ( ( ( ord_less_rat @ X3 @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X3 @ bot_bot_set_rat ) )
          = ( insert_rat @ X3 @ bot_bot_set_rat ) ) )
      & ( ~ ( ord_less_rat @ X3 @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X3 @ bot_bot_set_rat ) )
          = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
thf(fact_8426_Iio__Int__singleton,axiom,
    ! [X3: num,K2: num] :
      ( ( ( ord_less_num @ X3 @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X3 @ bot_bot_set_num ) )
          = ( insert_num @ X3 @ bot_bot_set_num ) ) )
      & ( ~ ( ord_less_num @ X3 @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X3 @ bot_bot_set_num ) )
          = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
thf(fact_8427_Iio__Int__singleton,axiom,
    ! [X3: int,K2: int] :
      ( ( ( ord_less_int @ X3 @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
          = ( insert_int @ X3 @ bot_bot_set_int ) ) )
      & ( ~ ( ord_less_int @ X3 @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
          = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
thf(fact_8428_Iio__Int__singleton,axiom,
    ! [X3: nat,K2: nat] :
      ( ( ( ord_less_nat @ X3 @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) )
      & ( ~ ( ord_less_nat @ X3 @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
          = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
thf(fact_8429_Iio__Int__singleton,axiom,
    ! [X3: real,K2: real] :
      ( ( ( ord_less_real @ X3 @ K2 )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
          = ( insert_real @ X3 @ bot_bot_set_real ) ) )
      & ( ~ ( ord_less_real @ X3 @ K2 )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
          = bot_bot_set_real ) ) ) ).

% Iio_Int_singleton
thf(fact_8430_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A4: set_nat,S5: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( S5
            = ( plus_plus_real @ S3
              @ ( groups6591440286371151544t_real
                @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
                @ A4 ) ) )
         => ( sums_real
            @ ^ [N3: nat] : ( if_real @ ( member_nat @ N3 @ A4 ) @ ( F @ N3 ) @ ( G @ N3 ) )
            @ S5 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8431_suminf__le__const,axiom,
    ! [F: nat > int,X3: int] :
      ( ( summable_int @ F )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8432_suminf__le__const,axiom,
    ! [F: nat > nat,X3: nat] :
      ( ( summable_nat @ F )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8433_suminf__le__const,axiom,
    ! [F: nat > real,X3: real] :
      ( ( summable_real @ F )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8434_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_8435_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_8436_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_8437_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_8438_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8439_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8440_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_8441_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N3: nat] : ( minus_minus_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8442_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N3: nat] : ( minus_minus_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8443_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N3: nat] : ( minus_minus_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F @ M2 ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_8444_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X3: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8445_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X3: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8446_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X3: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ X3 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8447_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8448_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_8449_sum__Un__nat,axiom,
    ! [A4: set_complex,B4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( sup_sup_set_complex @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F @ A4 ) @ ( groups5693394587270226106ex_nat @ F @ B4 ) ) @ ( groups5693394587270226106ex_nat @ F @ ( inf_inf_set_complex @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8450_sum__Un__nat,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B4 )
       => ( ( groups977919841031483927at_nat @ F @ ( sup_su6327502436637775413at_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups977919841031483927at_nat @ F @ A4 ) @ ( groups977919841031483927at_nat @ F @ B4 ) ) @ ( groups977919841031483927at_nat @ F @ ( inf_in2572325071724192079at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8451_sum__Un__nat,axiom,
    ! [A4: set_Pr4329608150637261639at_nat,B4: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( finite4343798906461161616at_nat @ A4 )
     => ( ( finite4343798906461161616at_nat @ B4 )
       => ( ( groups3860910324918113789at_nat @ F @ ( sup_su5525570899277871387at_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3860910324918113789at_nat @ F @ A4 ) @ ( groups3860910324918113789at_nat @ F @ B4 ) ) @ ( groups3860910324918113789at_nat @ F @ ( inf_in7913087082777306421at_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8452_sum__Un__nat,axiom,
    ! [A4: set_nat,B4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( groups3542108847815614940at_nat @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
          = ( minus_minus_nat @ ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ A4 ) @ ( groups3542108847815614940at_nat @ F @ B4 ) ) @ ( groups3542108847815614940at_nat @ F @ ( inf_inf_set_nat @ A4 @ B4 ) ) ) ) ) ) ).

% sum_Un_nat
thf(fact_8453_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_8454_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_8455_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_8456_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_8457_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_8458_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_8459_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_8460_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_8461_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_8462_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_8463_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_8464_suminf__split__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real @ F )
        = ( plus_plus_real
          @ ( suminf_real
            @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
          @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).

% suminf_split_initial_segment
thf(fact_8465_suminf__minus__initial__segment,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ K2 ) ) )
        = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) ) ) ) ).

% suminf_minus_initial_segment
thf(fact_8466_sum__less__suminf,axiom,
    ! [F: nat > int,N: nat] :
      ( ( summable_int @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_int @ zero_zero_int @ ( F @ M ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8467_sum__less__suminf,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( summable_nat @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8468_sum__less__suminf,axiom,
    ! [F: nat > real,N: nat] :
      ( ( summable_real @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_real @ zero_zero_real @ ( F @ M ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_8469_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_8470_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_8471_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_8472_lemma__termdiff1,axiom,
    ! [Z2: complex,H: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z2 @ P5 ) ) @ ( power_power_complex @ Z2 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_8473_lemma__termdiff1,axiom,
    ! [Z2: rat,H: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z2 @ P5 ) ) @ ( power_power_rat @ Z2 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_8474_lemma__termdiff1,axiom,
    ! [Z2: int,H: int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z2 @ P5 ) ) @ ( power_power_int @ Z2 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_8475_lemma__termdiff1,axiom,
    ! [Z2: real,H: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z2 @ P5 ) ) @ ( power_power_real @ Z2 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_8476_power__diff__sumr2,axiom,
    ! [X3: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N ) @ ( power_power_complex @ Y @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I4: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_complex @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8477_power__diff__sumr2,axiom,
    ! [X3: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N ) @ ( power_power_rat @ Y @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I4: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_rat @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8478_power__diff__sumr2,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ N ) @ ( power_power_int @ Y @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I4: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_int @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8479_power__diff__sumr2,axiom,
    ! [X3: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ N ) @ ( power_power_real @ Y @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I4: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I4 ) ) ) @ ( power_power_real @ X3 @ I4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_8480_diff__power__eq__sum,axiom,
    ! [X3: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8481_diff__power__eq__sum,axiom,
    ! [X3: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8482_diff__power__eq__sum,axiom,
    ! [X3: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X3 @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8483_diff__power__eq__sum,axiom,
    ! [X3: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X3 @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_8484_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_8485_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_8486_power__half__series,axiom,
    ( sums_real
    @ ^ [N3: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N3 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_8487_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > rat,K5: rat,K2: nat] :
      ( ! [P6: nat] :
          ( ( ord_less_nat @ P6 @ N )
         => ( ord_less_eq_rat @ ( F @ P6 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8488_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > int,K5: int,K2: nat] :
      ( ! [P6: nat] :
          ( ( ord_less_nat @ P6 @ N )
         => ( ord_less_eq_int @ ( F @ P6 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8489_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > nat,K5: nat,K2: nat] :
      ( ! [P6: nat] :
          ( ( ord_less_nat @ P6 @ N )
         => ( ord_less_eq_nat @ ( F @ P6 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8490_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F: nat > real,K5: real,K2: nat] :
      ( ! [P6: nat] :
          ( ( ord_less_nat @ P6 @ N )
         => ( ord_less_eq_real @ ( F @ P6 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_8491_sum__less__suminf2,axiom,
    ! [F: nat > int,N: nat,I: nat] :
      ( ( summable_int @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M ) ) )
       => ( ( 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_8492_sum__less__suminf2,axiom,
    ! [F: nat > nat,N: nat,I: nat] :
      ( ( summable_nat @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M ) ) )
       => ( ( 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_8493_sum__less__suminf2,axiom,
    ! [F: nat > real,N: nat,I: nat] :
      ( ( summable_real @ F )
     => ( ! [M: nat] :
            ( ( ord_less_eq_nat @ N @ M )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M ) ) )
       => ( ( 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_8494_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_8495_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_8496_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_8497_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_8498_sums__if_H,axiom,
    ! [G: nat > real,X3: real] :
      ( ( sums_real @ G @ X3 )
     => ( sums_real
        @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X3 ) ) ).

% sums_if'
thf(fact_8499_sums__if,axiom,
    ! [G: nat > real,X3: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X3 )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( F @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X3 @ Y ) ) ) ) ).

% sums_if
thf(fact_8500_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_8501_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_8502_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_8503_Sum__Icc__int,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ N )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X4: int] : X4
          @ ( set_or1266510415728281911st_int @ M2 @ N ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N @ ( plus_plus_int @ N @ one_one_int ) ) @ ( times_times_int @ M2 @ ( minus_minus_int @ M2 @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_8504_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K2: nat] :
      ( ( summable_real @ F )
     => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K2 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K2 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8505_mono__SucI1,axiom,
    ! [X6: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI1
thf(fact_8506_mono__SucI1,axiom,
    ! [X6: nat > set_nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% mono_SucI1
thf(fact_8507_mono__SucI1,axiom,
    ! [X6: nat > rat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI1
thf(fact_8508_mono__SucI1,axiom,
    ! [X6: nat > num] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI1
thf(fact_8509_mono__SucI1,axiom,
    ! [X6: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI1
thf(fact_8510_mono__SucI1,axiom,
    ! [X6: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI1
thf(fact_8511_mono__SucI2,axiom,
    ! [X6: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% mono_SucI2
thf(fact_8512_mono__SucI2,axiom,
    ! [X6: nat > set_nat] :
      ( ! [N2: nat] : ( ord_less_eq_set_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% mono_SucI2
thf(fact_8513_mono__SucI2,axiom,
    ! [X6: nat > rat] :
      ( ! [N2: nat] : ( ord_less_eq_rat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% mono_SucI2
thf(fact_8514_mono__SucI2,axiom,
    ! [X6: nat > num] :
      ( ! [N2: nat] : ( ord_less_eq_num @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% mono_SucI2
thf(fact_8515_mono__SucI2,axiom,
    ! [X6: nat > nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% mono_SucI2
thf(fact_8516_mono__SucI2,axiom,
    ! [X6: nat > int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% mono_SucI2
thf(fact_8517_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X8: nat > real] :
          ( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8518_monoseq__Suc,axiom,
    ( topolo7278393974255667507et_nat
    = ( ^ [X8: nat > set_nat] :
          ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_set_nat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8519_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X8: nat > rat] :
          ( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8520_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X8: nat > num] :
          ( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8521_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X8: nat > nat] :
          ( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8522_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X8: nat > int] :
          ( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
          | ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8523_monoI1,axiom,
    ! [X6: nat > real] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_real @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo6980174941875973593q_real @ X6 ) ) ).

% monoI1
thf(fact_8524_monoI1,axiom,
    ! [X6: nat > set_nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_set_nat @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo7278393974255667507et_nat @ X6 ) ) ).

% monoI1
thf(fact_8525_monoI1,axiom,
    ! [X6: nat > rat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_rat @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo4267028734544971653eq_rat @ X6 ) ) ).

% monoI1
thf(fact_8526_monoI1,axiom,
    ! [X6: nat > num] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_num @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo1459490580787246023eq_num @ X6 ) ) ).

% monoI1
thf(fact_8527_monoI1,axiom,
    ! [X6: nat > nat] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_nat @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo4902158794631467389eq_nat @ X6 ) ) ).

% monoI1
thf(fact_8528_monoI1,axiom,
    ! [X6: nat > int] :
      ( ! [M: nat,N2: nat] :
          ( ( ord_less_eq_nat @ M @ N2 )
         => ( ord_less_eq_int @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
     => ( topolo4899668324122417113eq_int @ X6 ) ) ).

% monoI1
thf(fact_8529_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8530_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8531_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ zero_zero_real @ M5 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8532_diffs__equiv,axiom,
    ! [C: nat > complex,X3: complex] :
      ( ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) )
     => ( sums_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N3 ) @ ( C @ N3 ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8533_diffs__equiv,axiom,
    ! [C: nat > real,X3: real] :
      ( ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) )
     => ( sums_real
        @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( C @ N3 ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) ) ) ) ).

% diffs_equiv
thf(fact_8534_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_8535_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [K: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K ) ) ) ).

% finite_nat_bounded
thf(fact_8536_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C4: nat > rat,N3: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) @ ( C4 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8537_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C4: nat > real,N3: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) @ ( C4 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8538_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C4: nat > int,N3: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) @ ( C4 @ ( suc @ N3 ) ) ) ) ) ).

% diffs_def
thf(fact_8539_termdiff__converges__all,axiom,
    ! [C: nat > complex,X3: complex] :
      ( ! [X5: complex] :
          ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) )
     => ( summable_complex
        @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8540_termdiff__converges__all,axiom,
    ! [C: nat > real,X3: real] :
      ( ! [X5: real] :
          ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) )
     => ( summable_real
        @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) ) ) ).

% termdiff_converges_all
thf(fact_8541_finite__transitivity__chain,axiom,
    ! [A4: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X5: real] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: real,Y4: real,Z3: real] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ A4 )
               => ? [Y6: real] :
                    ( ( member_real @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8542_finite__transitivity__chain,axiom,
    ! [A4: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X5: complex] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: complex,Y4: complex,Z3: complex] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ A4 )
               => ? [Y6: complex] :
                    ( ( member_complex @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8543_finite__transitivity__chain,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ! [X5: product_prod_nat_nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: product_prod_nat_nat,Y4: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X5 @ A4 )
               => ? [Y6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8544_finite__transitivity__chain,axiom,
    ! [A4: set_o,R: $o > $o > $o] :
      ( ( finite_finite_o @ A4 )
     => ( ! [X5: $o] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: $o,Y4: $o,Z3: $o] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ A4 )
               => ? [Y6: $o] :
                    ( ( member_o @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bot_set_o ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8545_finite__transitivity__chain,axiom,
    ! [A4: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X5: nat] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y4: nat,Z3: nat] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: nat] :
                ( ( member_nat @ X5 @ A4 )
               => ? [Y6: nat] :
                    ( ( member_nat @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8546_finite__transitivity__chain,axiom,
    ! [A4: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X5: int] :
            ~ ( R @ X5 @ X5 )
       => ( ! [X5: int,Y4: int,Z3: int] :
              ( ( R @ X5 @ Y4 )
             => ( ( R @ Y4 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ A4 )
               => ? [Y6: int] :
                    ( ( member_int @ Y6 @ A4 )
                    & ( R @ X5 @ Y6 ) ) )
           => ( A4 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_8547_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
              @ ^ [N3: nat] : ( times_times_real @ ( C @ N3 ) @ ( power_power_real @ X5 @ N3 ) ) ) )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( diffs_real @ C @ N3 ) @ ( power_power_real @ X3 @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8548_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
              @ ^ [N3: nat] : ( times_times_complex @ ( C @ N3 ) @ ( power_power_complex @ X5 @ N3 ) ) ) )
       => ( summable_complex
          @ ^ [N3: nat] : ( times_times_complex @ ( diffs_complex @ C @ N3 ) @ ( power_power_complex @ X3 @ N3 ) ) ) ) ) ).

% termdiff_converges
thf(fact_8549_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M5: nat] :
          ? [N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ N3 )
            & ( member_nat @ N3 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_8550_Maclaurin__cos__expansion2,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ T5 )
            & ( ord_less_real @ T5 @ X3 )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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_8551_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ? [T5: real] :
            ( ( ord_less_real @ X3 @ T5 )
            & ( ord_less_real @ T5 @ zero_zero_real )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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_8552_Maclaurin__cos__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
      & ( ( cos_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( cos_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T5 @ ( 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_8553_sin__paired,axiom,
    ! [X3: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ one_one_nat ) ) )
      @ ( sin_real @ X3 ) ) ).

% sin_paired
thf(fact_8554_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8555_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_8556_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_8557_of__real__power,axiom,
    ! [X3: real,N: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X3 @ N ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X3 ) @ N ) ) ).

% of_real_power
thf(fact_8558_of__real__power,axiom,
    ! [X3: real,N: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X3 @ N ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X3 ) @ N ) ) ).

% of_real_power
thf(fact_8559_of__real__add,axiom,
    ! [X3: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X3 @ Y ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_add
thf(fact_8560_of__real__add,axiom,
    ! [X3: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X3 @ Y ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_add
thf(fact_8561_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_8562_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_8563_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_8564_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_8565_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_8566_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_8567_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_8568_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_8569_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_8570_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8571_fact__2,axiom,
    ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8572_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8573_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8574_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_8575_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8576_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_8577_norm__of__real__add1,axiom,
    ! [X3: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8578_norm__of__real__add1,axiom,
    ! [X3: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8579_norm__of__real__addn,axiom,
    ! [X3: real,B: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( numeral_numeral_real @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8580_norm__of__real__addn,axiom,
    ! [X3: real,B: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( numera6690914467698888265omplex @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8581_cos__of__real__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_of_real_pi_half
thf(fact_8582_cos__of__real__pi__half,axiom,
    ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = zero_zero_complex ) ).

% cos_of_real_pi_half
thf(fact_8583_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_8584_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_8585_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_8586_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_8587_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_8588_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_8589_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_8590_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_8591_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_8592_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_8593_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_mono
thf(fact_8594_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_mono
thf(fact_8595_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_mono
thf(fact_8596_fact__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono
thf(fact_8597_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) ) ) ).

% fact_dvd
thf(fact_8598_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M2 ) ) ) ).

% fact_dvd
thf(fact_8599_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ M2 ) ) ) ).

% fact_dvd
thf(fact_8600_fact__dvd,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) ) ) ).

% fact_dvd
thf(fact_8601_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ N ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8602_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ N ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8603_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ N ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8604_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8605_fact__fact__dvd__fact,axiom,
    ! [K2: nat,N: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K2 @ N ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8606_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N ) @ ( semiri1406184849735516958ct_int @ M2 ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_8607_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N ) @ ( semiri3624122377584611663nteger @ M2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_8608_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ M2 ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_8609_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_8610_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_8611_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_8612_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_8613_norm__less__p1,axiom,
    ! [X3: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X3 ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8614_norm__less__p1,axiom,
    ! [X3: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X3 ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8615_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K2 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri3624122377584611663nteger @ N ) ) ) ).

% choose_dvd
thf(fact_8616_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% choose_dvd
thf(fact_8617_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% choose_dvd
thf(fact_8618_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% choose_dvd
thf(fact_8619_choose__dvd,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% choose_dvd
thf(fact_8620_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8621_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri773545260158071498ct_rat @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8622_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_int @ ( numeral_numeral_int @ K2 ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8623_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_real @ ( numeral_numeral_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8624_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8625_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_8626_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X4: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).

% cos_sin_eq
thf(fact_8627_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X4: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).

% cos_sin_eq
thf(fact_8628_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X4: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).

% sin_cos_eq
thf(fact_8629_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X4: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X4 ) ) ) ) ).

% sin_cos_eq
thf(fact_8630_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > literal > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_literal ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_literal ) ) ) ) ).

% Maclaurin_zero
thf(fact_8631_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > real > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8632_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > rat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8633_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > nat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8634_Maclaurin__zero,axiom,
    ! [X3: real,N: nat,Diff: nat > int > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8635_Maclaurin__lemma,axiom,
    ! [H: real,F: real > real,J: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ? [B6: real] :
          ( ( F @ H )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B6 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8636_minus__sin__cos__eq,axiom,
    ! [X3: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X3 ) )
      = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8637_minus__sin__cos__eq,axiom,
    ! [X3: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X3 ) )
      = ( cos_complex @ ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8638_Maclaurin__exp__le,axiom,
    ! [X3: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
      & ( ( exp_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8639_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_8640_cos__paired,axiom,
    ! [X3: real] :
      ( sums_real
      @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) @ ( power_power_real @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) )
      @ ( cos_real @ X3 ) ) ).

% cos_paired
thf(fact_8641_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X4: real] : ( ln_ln_real @ ( plus_plus_real @ X4 @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_8642_Maclaurin__exp__lt,axiom,
    ! [X3: real,N: nat] :
      ( ( X3 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
            & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
            & ( ( exp_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M5 ) @ ( semiri2265585572941072030t_real @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8643_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ? [T5: real] :
            ( ( ord_less_real @ zero_zero_real @ T5 )
            & ( ord_less_real @ T5 @ X3 )
            & ( ( sin_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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_8644_Maclaurin__sin__expansion4,axiom,
    ! [X3: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ? [T5: real] :
          ( ( ord_less_real @ zero_zero_real @ T5 )
          & ( ord_less_eq_real @ T5 @ X3 )
          & ( ( sin_real @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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_8645_Maclaurin__sin__expansion2,axiom,
    ! [X3: real,N: nat] :
    ? [T5: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
      & ( ( sin_real @ X3 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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_8646_Maclaurin__sin__expansion,axiom,
    ! [X3: real,N: nat] :
    ? [T5: real] :
      ( ( sin_real @ X3 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T5 @ ( 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_8647_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N3 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8648_fact__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_mono_nat
thf(fact_8649_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_8650_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_8651_dvd__fact,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( dvd_dvd_nat @ M2 @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% dvd_fact
thf(fact_8652_fact__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ ( suc @ M2 ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ N ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8653_fact__div__fact__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R2 ) ) ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_8654_sin__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( sin_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( cos_coeff @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% sin_coeff_Suc
thf(fact_8655_cos__coeff__Suc,axiom,
    ! [N: nat] :
      ( ( cos_coeff @ ( suc @ N ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ).

% cos_coeff_Suc
thf(fact_8656_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( semiri1314217659103216013at_int @ N ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8657_pochhammer__double,axiom,
    ! [Z2: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_8658_pochhammer__double,axiom,
    ! [Z2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_8659_pochhammer__double,axiom,
    ! [Z2: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_8660_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N3: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I4: complex] : ( plus_plus_complex @ I4 @ one_one_complex )
          @ N3
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_8661_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N3: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I4: rat] : ( plus_plus_rat @ I4 @ one_one_rat )
          @ N3
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8662_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N3: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I4: real] : ( plus_plus_real @ I4 @ one_one_real )
          @ N3
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8663_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I4: int] : ( plus_plus_int @ I4 @ one_one_int )
          @ N3
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8664_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ one_one_nat )
          @ N3
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8665_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M2 ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8666_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M2 ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8667_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8668_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_8669_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_8670_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K2 ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_8671_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K2 ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_8672_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_8673_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K2 ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_8674_floor__numeral,axiom,
    ! [V2: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V2 ) )
      = ( numeral_numeral_int @ V2 ) ) ).

% floor_numeral
thf(fact_8675_floor__numeral,axiom,
    ! [V2: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V2 ) )
      = ( numeral_numeral_int @ V2 ) ) ).

% floor_numeral
thf(fact_8676_binomial__0__Suc,axiom,
    ! [K2: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8677_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_8678_binomial__Suc__Suc,axiom,
    ! [N: nat,K2: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
      = ( plus_plus_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).

% binomial_Suc_Suc
thf(fact_8679_zero__less__binomial__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) )
      = ( ord_less_eq_nat @ K2 @ N ) ) ).

% zero_less_binomial_iff
thf(fact_8680_zero__le__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X3 ) ) ).

% zero_le_floor
thf(fact_8681_zero__le__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% zero_le_floor
thf(fact_8682_numeral__le__floor,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V2 ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V2 ) @ X3 ) ) ).

% numeral_le_floor
thf(fact_8683_numeral__le__floor,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V2 ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V2 ) @ X3 ) ) ).

% numeral_le_floor
thf(fact_8684_zero__less__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).

% zero_less_floor
thf(fact_8685_zero__less__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).

% zero_less_floor
thf(fact_8686_floor__less__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ V2 ) ) ) ).

% floor_less_numeral
thf(fact_8687_floor__less__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_real @ X3 @ ( numeral_numeral_real @ V2 ) ) ) ).

% floor_less_numeral
thf(fact_8688_one__le__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).

% one_le_floor
thf(fact_8689_one__le__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).

% one_le_floor
thf(fact_8690_floor__neg__numeral,axiom,
    ! [V2: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ).

% floor_neg_numeral
thf(fact_8691_floor__neg__numeral,axiom,
    ! [V2: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) ) ).

% floor_neg_numeral
thf(fact_8692_floor__diff__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X3 @ ( numeral_numeral_rat @ V2 ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% floor_diff_numeral
thf(fact_8693_floor__diff__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X3 @ ( numeral_numeral_real @ V2 ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V2 ) ) ) ).

% floor_diff_numeral
thf(fact_8694_floor__numeral__power,axiom,
    ! [X3: num,N: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ).

% floor_numeral_power
thf(fact_8695_floor__numeral__power,axiom,
    ! [X3: num,N: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N ) )
      = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ).

% floor_numeral_power
thf(fact_8696_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_8697_numeral__less__floor,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V2 ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V2 ) @ one_one_rat ) @ X3 ) ) ).

% numeral_less_floor
thf(fact_8698_numeral__less__floor,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V2 ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V2 ) @ one_one_real ) @ X3 ) ) ).

% numeral_less_floor
thf(fact_8699_floor__le__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V2 ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_8700_floor__le__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V2 ) )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( numeral_numeral_real @ V2 ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_8701_one__less__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) ) ).

% one_less_floor
thf(fact_8702_one__less__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ).

% one_less_floor
thf(fact_8703_floor__le__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
      = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8704_floor__le__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
      = ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8705_neg__numeral__le__floor,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ X3 ) ) ).

% neg_numeral_le_floor
thf(fact_8706_neg__numeral__le__floor,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ X3 ) ) ).

% neg_numeral_le_floor
thf(fact_8707_floor__less__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8708_floor__less__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8709_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_8710_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_8711_neg__numeral__less__floor,axiom,
    ! [V2: num,X3: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ one_one_rat ) @ X3 ) ) ).

% neg_numeral_less_floor
thf(fact_8712_neg__numeral__less__floor,axiom,
    ! [V2: num,X3: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ one_one_real ) @ X3 ) ) ).

% neg_numeral_less_floor
thf(fact_8713_floor__le__neg__numeral,axiom,
    ! [X3: rat,V2: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_8714_floor__le__neg__numeral,axiom,
    ! [X3: real,V2: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_8715_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_8716_floor__mono,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_8717_floor__mono,axiom,
    ! [X3: real,Y: real] :
      ( ( ord_less_eq_real @ X3 @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_8718_of__int__floor__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 ) ).

% of_int_floor_le
thf(fact_8719_of__int__floor__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 ) ).

% of_int_floor_le
thf(fact_8720_Suc__times__binomial__eq,axiom,
    ! [N: nat,K2: nat] :
      ( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).

% Suc_times_binomial_eq
thf(fact_8721_Suc__times__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
      = ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).

% Suc_times_binomial
thf(fact_8722_binomial__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( binomial @ N @ K2 )
        = ( binomial @ N @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% binomial_symmetric
thf(fact_8723_choose__mult__lemma,axiom,
    ! [M2: nat,R2: nat,K2: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus_nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K2 ) @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M2 @ R2 ) @ M2 ) ) ) ).

% choose_mult_lemma
thf(fact_8724_binomial__le__pow,axiom,
    ! [R2: nat,N: nat] :
      ( ( ord_less_eq_nat @ R2 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).

% binomial_le_pow
thf(fact_8725_pochhammer__eq__0__mono,axiom,
    ! [A: real,N: nat,M2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ M2 )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8726_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N: nat,M2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ M2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8727_pochhammer__neq__0__mono,axiom,
    ! [A: real,M2: nat,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s7457072308508201937r_real @ A @ N )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8728_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M2: nat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M2 )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s4028243227959126397er_rat @ A @ N )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8729_zero__less__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K2 ) ) ) ).

% zero_less_binomial
thf(fact_8730_le__floor__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).

% le_floor_iff
thf(fact_8731_le__floor__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).

% le_floor_iff
thf(fact_8732_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_8733_binomial__Suc__Suc__eq__times,axiom,
    ! [N: nat,K2: nat] :
      ( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) @ ( suc @ K2 ) ) ) ).

% binomial_Suc_Suc_eq_times
thf(fact_8734_le__floor__add,axiom,
    ! [X3: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y ) ) ) ).

% le_floor_add
thf(fact_8735_le__floor__add,axiom,
    ! [X3: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y ) ) ) ).

% le_floor_add
thf(fact_8736_int__add__floor,axiom,
    ! [Z2: int,X3: rat] :
      ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ) ).

% int_add_floor
thf(fact_8737_int__add__floor,axiom,
    ! [Z2: int,X3: real] :
      ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ) ).

% int_add_floor
thf(fact_8738_floor__add__int,axiom,
    ! [X3: rat,Z2: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).

% floor_add_int
thf(fact_8739_floor__add__int,axiom,
    ! [X3: real,Z2: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).

% floor_add_int
thf(fact_8740_choose__mult,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( times_times_nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
          = ( times_times_nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus_nat @ N @ K2 ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ) ).

% choose_mult
thf(fact_8741_binomial__fact__lemma,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
        = ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% binomial_fact_lemma
thf(fact_8742_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A6: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A6 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8743_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A6: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A6 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8744_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A6: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A6 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8745_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A6: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A6 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8746_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A6: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A6 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8747_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A6: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A6 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8748_floor__power,axiom,
    ! [X3: rat,N: nat] :
      ( ( X3
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X3 @ N ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X3 ) @ N ) ) ) ).

% floor_power
thf(fact_8749_floor__power,axiom,
    ! [X3: real,N: nat] :
      ( ( X3
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X3 @ N ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X3 ) @ N ) ) ) ).

% floor_power
thf(fact_8750_gbinomial__Suc__Suc,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ A @ K2 ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8751_gbinomial__Suc__Suc,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( plus_plus_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8752_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8753_gbinomial__of__nat__symmetric,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ K2 )
        = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N ) @ ( minus_minus_nat @ N @ K2 ) ) ) ) ).

% gbinomial_of_nat_symmetric
thf(fact_8754_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_8755_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_8756_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_8757_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_8758_one__add__floor,axiom,
    ! [X3: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_8759_one__add__floor,axiom,
    ! [X3: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_8760_binomial__absorption,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_nat @ ( suc @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ).

% binomial_absorption
thf(fact_8761_binomial__altdef__nat,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( binomial @ N @ K2 )
        = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_altdef_nat
thf(fact_8762_gbinomial__addition__formula,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K2 ) )
      = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_8763_gbinomial__addition__formula,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K2 ) )
      = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_8764_gbinomial__addition__formula,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_addition_formula
thf(fact_8765_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K2 ) @ A )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8766_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K2 ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8767_gbinomial__mult__1_H,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8768_gbinomial__mult__1_H,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ A )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8769_gbinomial__mult__1,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K2 ) @ ( gbinomial_rat @ A @ K2 ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8770_gbinomial__mult__1,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ A @ ( gbinomial_real @ A @ K2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K2 ) @ ( gbinomial_real @ A @ K2 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8771_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_8772_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_8773_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_8774_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_8775_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_8776_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_8777_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_8778_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_8779_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_8780_pochhammer__rec_H,axiom,
    ! [Z2: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8781_pochhammer__rec_H,axiom,
    ! [Z2: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8782_pochhammer__rec_H,axiom,
    ! [Z2: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8783_pochhammer__rec_H,axiom,
    ! [Z2: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_8784_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
       != zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8785_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8786_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8787_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
       != zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8788_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
       != zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8789_pochhammer__product_H,axiom,
    ! [Z2: rat,N: nat,M2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_8790_pochhammer__product_H,axiom,
    ! [Z2: real,N: nat,M2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_8791_pochhammer__product_H,axiom,
    ! [Z2: int,N: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_8792_pochhammer__product_H,axiom,
    ! [Z2: nat,N: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_8793_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I4 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I4 ) @ one_one_rat ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_8794_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I4: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I4 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I4 ) @ one_one_real ) ) )
           => ( P @ I4 ) ) ) ) ).

% floor_split
thf(fact_8795_floor__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X3 )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X3 )
        & ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_8796_floor__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X3 )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X3 )
        & ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_8797_floor__unique,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
     => ( ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X3 )
          = Z2 ) ) ) ).

% floor_unique
thf(fact_8798_floor__unique,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = Z2 ) ) ) ).

% floor_unique
thf(fact_8799_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri5074537144036343181t_real @ K2 ) ) @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8800_binomial__ge__n__over__k__pow__k,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri681578069525770553at_rat @ K2 ) ) @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8801_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8802_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8803_less__floor__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).

% less_floor_iff
thf(fact_8804_less__floor__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).

% less_floor_iff
thf(fact_8805_binomial__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_mono
thf(fact_8806_binomial__maximum_H,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% binomial_maximum'
thf(fact_8807_floor__le__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_8808_floor__le__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_8809_binomial__maximum,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% binomial_maximum
thf(fact_8810_binomial__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K2 )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_eq_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).

% binomial_antimono
thf(fact_8811_binomial__le__pow2,axiom,
    ! [N: nat,K2: nat] : ( ord_less_eq_nat @ ( binomial @ N @ K2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).

% binomial_le_pow2
thf(fact_8812_floor__correct,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8813_floor__correct,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8814_choose__reduce__nat,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ( binomial @ N @ K2 )
          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ) ).

% choose_reduce_nat
thf(fact_8815_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_8816_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_8817_Suc__times__gbinomial,axiom,
    ! [K2: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% Suc_times_gbinomial
thf(fact_8818_gbinomial__absorption,axiom,
    ! [K2: nat,A: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) @ ( gbinomial_complex @ A @ ( suc @ K2 ) ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_8819_gbinomial__absorption,axiom,
    ! [K2: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) @ ( gbinomial_rat @ A @ ( suc @ K2 ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_8820_gbinomial__absorption,axiom,
    ! [K2: nat,A: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) @ ( gbinomial_real @ A @ ( suc @ K2 ) ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_absorption
thf(fact_8821_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: rat] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M2 ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M2 ) @ K2 ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8822_gbinomial__trinomial__revision,axiom,
    ! [K2: nat,M2: nat,A: real] :
      ( ( ord_less_eq_nat @ K2 @ M2 )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M2 ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M2 ) @ K2 ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( minus_minus_nat @ M2 @ K2 ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8823_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z2 @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_8824_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s7457072308508201937r_real @ Z2 @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_8825_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4660882817536571857er_int @ Z2 @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_8826_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z2 @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_8827_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8828_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_8829_binomial__less__binomial__Suc,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).

% binomial_less_binomial_Suc
thf(fact_8830_binomial__strict__mono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N )
       => ( ord_less_nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K6 ) ) ) ) ).

% binomial_strict_mono
thf(fact_8831_binomial__strict__antimono,axiom,
    ! [K2: nat,K6: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ K6 )
     => ( ( ord_less_eq_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) )
       => ( ( ord_less_eq_nat @ K6 @ N )
         => ( ord_less_nat @ ( binomial @ N @ K6 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).

% binomial_strict_antimono
thf(fact_8832_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_8833_binomial__addition__formula,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( binomial @ N @ ( suc @ K2 ) )
        = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ) ).

% binomial_addition_formula
thf(fact_8834_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8835_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8836_binomial__fact,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8837_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K2 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K2 ) ) )
        = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8838_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K2 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K2 ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8839_fact__binomial,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K2 @ N )
     => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K2 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K2 ) ) )
        = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ K2 ) ) ) ) ) ).

% fact_binomial
thf(fact_8840_gbinomial__factors,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) @ ( gbinomial_complex @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_8841_gbinomial__factors,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) @ ( gbinomial_rat @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_8842_gbinomial__factors,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) @ ( gbinomial_real @ A @ K2 ) ) ) ).

% gbinomial_factors
thf(fact_8843_gbinomial__rec,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K2 ) )
      = ( times_times_complex @ ( gbinomial_complex @ A @ K2 ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8844_gbinomial__rec,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K2 ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K2 ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8845_gbinomial__rec,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K2 ) )
      = ( times_times_real @ ( gbinomial_real @ A @ K2 ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K2 ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8846_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ one_one_complex ) @ K2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8847_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ one_one_rat ) @ K2 ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8848_gbinomial__index__swap,axiom,
    ! [K2: nat,N: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ one_one_real ) @ K2 ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ N ) ) ) ).

% gbinomial_index_swap
thf(fact_8849_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A6: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A6 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8850_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A6: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A6 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8851_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A6: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A6 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8852_pochhammer__absorb__comp,axiom,
    ! [R2: complex,K2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K2 ) )
      = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8853_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K2 ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K2 ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8854_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K2 ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8855_pochhammer__absorb__comp,axiom,
    ! [R2: real,K2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K2 ) )
      = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8856_pochhammer__absorb__comp,axiom,
    ! [R2: int,K2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K2 ) )
      = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8857_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8858_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_8859_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_8860_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_8861_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_8862_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_8863_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_8864_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_8865_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X4: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8866_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X4: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8867_gbinomial__minus,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_8868_gbinomial__minus,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_8869_gbinomial__minus,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_minus
thf(fact_8870_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_complex @ A @ K2 )
        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8871_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_real @ A @ K2 )
        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8872_gbinomial__reduce__nat,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8873_pochhammer__minus_H,axiom,
    ! [B: complex,K2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_8874_pochhammer__minus_H,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_8875_pochhammer__minus_H,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_8876_pochhammer__minus_H,axiom,
    ! [B: real,K2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_8877_pochhammer__minus_H,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 ) ) ) ).

% pochhammer_minus'
thf(fact_8878_pochhammer__minus,axiom,
    ! [B: complex,K2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K2 ) ) @ one_one_complex ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_8879_pochhammer__minus,axiom,
    ! [B: code_integer,K2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K2 ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K2 ) ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_8880_pochhammer__minus,axiom,
    ! [B: rat,K2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K2 ) ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_8881_pochhammer__minus,axiom,
    ! [B: real,K2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K2 ) ) @ one_one_real ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_8882_pochhammer__minus,axiom,
    ! [B: int,K2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K2 ) ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_minus
thf(fact_8883_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8884_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8885_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K2 )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( plus_plus_nat @ K2 @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8886_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_8887_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_8888_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_8889_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_8890_floor__log__nat__eq__if,axiom,
    ! [B: nat,N: nat,K2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N ) @ K2 )
     => ( ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8891_binomial__code,axiom,
    ( binomial
    = ( ^ [N3: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N3 @ ( minus_minus_nat @ N3 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N3 @ K3 ) @ one_one_nat ) @ N3 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8892_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( 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_8893_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8894_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8895_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A6: complex,N3: nat] :
          ( if_complex @ ( N3 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A6 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_8896_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A6: rat,N3: nat] :
          ( if_rat @ ( N3 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A6 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8897_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A6: real,N3: nat] :
          ( if_real @ ( N3 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A6 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_8898_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A6: int,N3: nat] :
          ( if_int @ ( N3 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A6 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_8899_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A6: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A6 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N3 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_8900_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8901_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8902_gbinomial__partial__row__sum,axiom,
    ! [A: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M2 @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8903_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_8904_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_8905_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_8906_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_8907_atMost__iff,axiom,
    ! [I: real,K2: real] :
      ( ( member_real @ I @ ( set_ord_atMost_real @ K2 ) )
      = ( ord_less_eq_real @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8908_atMost__iff,axiom,
    ! [I: $o,K2: $o] :
      ( ( member_o @ I @ ( set_ord_atMost_o @ K2 ) )
      = ( ord_less_eq_o @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8909_atMost__iff,axiom,
    ! [I: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K2 ) )
      = ( ord_less_eq_set_nat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8910_atMost__iff,axiom,
    ! [I: rat,K2: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K2 ) )
      = ( ord_less_eq_rat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8911_atMost__iff,axiom,
    ! [I: num,K2: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K2 ) )
      = ( ord_less_eq_num @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8912_atMost__iff,axiom,
    ! [I: int,K2: int] :
      ( ( member_int @ I @ ( set_ord_atMost_int @ K2 ) )
      = ( ord_less_eq_int @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8913_atMost__iff,axiom,
    ! [I: nat,K2: nat] :
      ( ( member_nat @ I @ ( set_ord_atMost_nat @ K2 ) )
      = ( ord_less_eq_nat @ I @ K2 ) ) ).

% atMost_iff
thf(fact_8914_prod_Oempty,axiom,
    ! [G: $o > complex] :
      ( ( groups4859619685533338977omplex @ G @ bot_bot_set_o )
      = one_one_complex ) ).

% prod.empty
thf(fact_8915_prod_Oempty,axiom,
    ! [G: $o > real] :
      ( ( groups234877984723959775o_real @ G @ bot_bot_set_o )
      = one_one_real ) ).

% prod.empty
thf(fact_8916_prod_Oempty,axiom,
    ! [G: $o > rat] :
      ( ( groups2869687844427037835_o_rat @ G @ bot_bot_set_o )
      = one_one_rat ) ).

% prod.empty
thf(fact_8917_prod_Oempty,axiom,
    ! [G: $o > nat] :
      ( ( groups3504817904513533571_o_nat @ G @ bot_bot_set_o )
      = one_one_nat ) ).

% prod.empty
thf(fact_8918_prod_Oempty,axiom,
    ! [G: $o > int] :
      ( ( groups3502327434004483295_o_int @ G @ bot_bot_set_o )
      = one_one_int ) ).

% prod.empty
thf(fact_8919_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_8920_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_8921_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_8922_prod_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
      = one_one_complex ) ).

% prod.empty
thf(fact_8923_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_8924_atMost__subset__iff,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X3 ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X3 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8925_atMost__subset__iff,axiom,
    ! [X3: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X3 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8926_atMost__subset__iff,axiom,
    ! [X3: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X3 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8927_atMost__subset__iff,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X3 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8928_atMost__subset__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X3 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8929_prod_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X3 @ A4 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A4 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8930_prod_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X3 @ A4 )
       => ( ( groups234877984723959775o_real @ G @ ( insert_o @ X3 @ A4 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups234877984723959775o_real @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8931_prod_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X3 @ A4 )
       => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A4 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8932_prod_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X3 @ A4 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A4 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8933_prod_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X3 @ A4 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A4 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8934_prod_Oinsert,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X3 @ A4 )
       => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A4 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8935_prod_Oinsert,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ~ ( member_o @ X3 @ A4 )
       => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8936_prod_Oinsert,axiom,
    ! [A4: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X3 @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8937_prod_Oinsert,axiom,
    ! [A4: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X3 @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8938_prod_Oinsert,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X3 @ A4 )
       => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_8939_Icc__subset__Iic__iff,axiom,
    ! [L: set_nat,H: set_nat,H3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ L @ H ) @ ( set_or4236626031148496127et_nat @ H3 ) )
      = ( ~ ( ord_less_eq_set_nat @ L @ H )
        | ( ord_less_eq_set_nat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8940_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H )
        | ( ord_less_eq_rat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8941_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H )
        | ( ord_less_eq_num @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8942_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H )
        | ( ord_less_eq_nat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8943_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H )
        | ( ord_less_eq_int @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8944_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L @ H )
        | ( ord_less_eq_real @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8945_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_8946_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_8947_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_8948_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_8949_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_8950_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_8951_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_8952_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_8953_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_8954_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_8955_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_8956_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_8957_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8958_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8959_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8960_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8961_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8962_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8963_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: $o] :
      ( bot_bot_set_o
     != ( set_ord_atMost_o @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_8964_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: int] :
      ( bot_bot_set_int
     != ( set_ord_atMost_int @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_8965_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: nat] :
      ( bot_bot_set_nat
     != ( set_ord_atMost_nat @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_8966_prod__power__distrib,axiom,
    ! [F: nat > nat,A4: set_nat,N: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ N )
      = ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( power_power_nat @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8967_prod__power__distrib,axiom,
    ! [F: nat > int,A4: set_nat,N: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A4 ) @ N )
      = ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( power_power_int @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8968_prod__power__distrib,axiom,
    ! [F: int > int,A4: set_int,N: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ N )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( power_power_int @ ( F @ X4 ) @ N )
        @ A4 ) ) ).

% prod_power_distrib
thf(fact_8969_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_8970_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_8971_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_8972_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_8973_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( groups708209901874060359at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [J3: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_8974_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( groups705719431365010083at_int @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [J3: nat] :
            ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_8975_atMost__def,axiom,
    ( set_or4236626031148496127et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_eq_set_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8976_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_eq_rat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8977_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8978_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8979_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8980_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_8981_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_8982_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_8983_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_8984_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_8985_prod__mono,axiom,
    ! [A4: set_o,F: $o > real,G: $o > real] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
            & ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A4 ) @ ( groups234877984723959775o_real @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8986_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_8987_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_8988_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_8989_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_8990_prod__mono,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ! [I3: $o] :
          ( ( member_o @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
            & ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ).

% prod_mono
thf(fact_8991_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_8992_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_8993_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_8994_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_8995_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups234877984723959775o_real @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_8996_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_8997_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_8998_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_8999_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_9000_prod__ge__1,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) ) ) ).

% prod_ge_1
thf(fact_9001_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_9002_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_9003_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_9004_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_9005_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_9006_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_9007_lessThan__Suc__atMost,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( set_ord_atMost_nat @ K2 ) ) ).

% lessThan_Suc_atMost
thf(fact_9008_atMost__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K2 ) )
      = ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).

% atMost_Suc
thf(fact_9009_not__Iic__le__Icc,axiom,
    ! [H: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_9010_not__Iic__le__Icc,axiom,
    ! [H: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_9011_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_9012_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_9013_power__sum,axiom,
    ! [C: real,F: nat > nat,A4: set_nat] :
      ( ( power_power_real @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups129246275422532515t_real
        @ ^ [A6: nat] : ( power_power_real @ C @ ( F @ A6 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_9014_power__sum,axiom,
    ! [C: complex,F: nat > nat,A4: set_nat] :
      ( ( power_power_complex @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups6464643781859351333omplex
        @ ^ [A6: nat] : ( power_power_complex @ C @ ( F @ A6 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_9015_power__sum,axiom,
    ! [C: nat,F: nat > nat,A4: set_nat] :
      ( ( power_power_nat @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups708209901874060359at_nat
        @ ^ [A6: nat] : ( power_power_nat @ C @ ( F @ A6 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_9016_power__sum,axiom,
    ! [C: int,F: nat > nat,A4: set_nat] :
      ( ( power_power_int @ C @ ( groups3542108847815614940at_nat @ F @ A4 ) )
      = ( groups705719431365010083at_int
        @ ^ [A6: nat] : ( power_power_int @ C @ ( F @ A6 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_9017_power__sum,axiom,
    ! [C: int,F: int > nat,A4: set_int] :
      ( ( power_power_int @ C @ ( groups4541462559716669496nt_nat @ F @ A4 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [A6: int] : ( power_power_int @ C @ ( F @ A6 ) )
        @ A4 ) ) ).

% power_sum
thf(fact_9018_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_9019_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > int,M2: nat,K2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( plus_plus_nat @ I4 @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_9020_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S6: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S6 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9021_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_9022_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_9023_prod__le__1,axiom,
    ! [A4: set_o,F: $o > real] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
            & ( ord_less_eq_real @ ( F @ X5 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_9024_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_9025_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_9026_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_9027_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_9028_prod__le__1,axiom,
    ! [A4: set_o,F: $o > rat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) )
            & ( ord_less_eq_rat @ ( F @ X5 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_9029_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_9030_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_9031_prod_Oinsert__if,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X3 @ A4 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A4 ) )
            = ( groups1681761925125756287l_real @ G @ A4 ) ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A4 ) )
            = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9032_prod_Oinsert__if,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X3 @ A4 )
         => ( ( groups234877984723959775o_real @ G @ ( insert_o @ X3 @ A4 ) )
            = ( groups234877984723959775o_real @ G @ A4 ) ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ( groups234877984723959775o_real @ G @ ( insert_o @ X3 @ A4 ) )
            = ( times_times_real @ ( G @ X3 ) @ ( groups234877984723959775o_real @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9033_prod_Oinsert__if,axiom,
    ! [A4: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X3 @ A4 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A4 ) )
            = ( groups2316167850115554303t_real @ G @ A4 ) ) )
        & ( ~ ( member_int @ X3 @ A4 )
         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A4 ) )
            = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9034_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X3: nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X3 @ A4 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( groups129246275422532515t_real @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X3 @ A4 )
         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9035_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X3 @ A4 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( groups766887009212190081x_real @ G @ A4 ) ) )
        & ( ~ ( member_complex @ X3 @ A4 )
         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9036_prod_Oinsert__if,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X3 @ A4 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( groups4061424788464935467al_rat @ G @ A4 ) ) )
        & ( ~ ( member_real @ X3 @ A4 )
         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A4 ) )
            = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9037_prod_Oinsert__if,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( member_o @ X3 @ A4 )
         => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
            = ( groups2869687844427037835_o_rat @ G @ A4 ) ) )
        & ( ~ ( member_o @ X3 @ A4 )
         => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
            = ( times_times_rat @ ( G @ X3 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9038_prod_Oinsert__if,axiom,
    ! [A4: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X3 @ A4 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
            = ( groups1072433553688619179nt_rat @ G @ A4 ) ) )
        & ( ~ ( member_int @ X3 @ A4 )
         => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
            = ( times_times_rat @ ( G @ X3 ) @ ( groups1072433553688619179nt_rat @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9039_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X3 @ A4 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( groups73079841787564623at_rat @ G @ A4 ) ) )
        & ( ~ ( member_nat @ X3 @ A4 )
         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
            = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9040_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X3 @ A4 )
         => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( groups225925009352817453ex_rat @ G @ A4 ) ) )
        & ( ~ ( member_complex @ X3 @ A4 )
         => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
            = ( times_times_rat @ ( G @ X3 ) @ ( groups225925009352817453ex_rat @ G @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_9041_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups4696554848551431203al_nat @ F @ A4 ) @ ( groups4696554848551431203al_nat @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9042_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > nat,G: $o > nat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups3504817904513533571_o_nat @ F @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9043_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups1707563613775114915nt_nat @ F @ A4 ) @ ( groups1707563613775114915nt_nat @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9044_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A4 )
             => ( dvd_dvd_nat @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9045_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: real > int,G: real > int] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups4694064378042380927al_int @ F @ A4 ) @ ( groups4694064378042380927al_int @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9046_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > int,G: $o > int] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups3502327434004483295_o_int @ F @ A4 ) @ ( groups3502327434004483295_o_int @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9047_prod__dvd__prod__subset2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ A4 )
             => ( dvd_dvd_int @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9048_prod__dvd__prod__subset2,axiom,
    ! [B4: set_real,A4: set_real,F: real > code_integer,G: real > code_integer] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups6225526099057966256nteger @ F @ A4 ) @ ( groups6225526099057966256nteger @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9049_prod__dvd__prod__subset2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > code_integer,G: $o > code_integer] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [A3: $o] :
              ( ( member_o @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups7694694392188491536nteger @ F @ A4 ) @ ( groups7694694392188491536nteger @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9050_prod__dvd__prod__subset2,axiom,
    ! [B4: set_int,A4: set_int,F: int > code_integer,G: int > code_integer] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ A4 )
             => ( dvd_dvd_Code_integer @ ( F @ A3 ) @ ( G @ A3 ) ) )
         => ( dvd_dvd_Code_integer @ ( groups3827104343326376752nteger @ F @ A4 ) @ ( groups3827104343326376752nteger @ G @ B4 ) ) ) ) ) ).

% prod_dvd_prod_subset2
thf(fact_9051_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( dvd_dvd_nat @ ( groups861055069439313189ex_nat @ F @ A4 ) @ ( groups861055069439313189ex_nat @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9052_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( dvd_dvd_nat @ ( groups4077766827762148844at_nat @ F @ A4 ) @ ( groups4077766827762148844at_nat @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9053_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( dvd_dvd_int @ ( groups858564598930262913ex_int @ F @ A4 ) @ ( groups858564598930262913ex_int @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9054_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( dvd_dvd_int @ ( groups4075276357253098568at_int @ F @ A4 ) @ ( groups4075276357253098568at_int @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9055_prod__dvd__prod__subset,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( dvd_dvd_Code_integer @ ( groups8682486955453173170nteger @ F @ A4 ) @ ( groups8682486955453173170nteger @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9056_prod__dvd__prod__subset,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > code_integer] :
      ( ( finite6177210948735845034at_nat @ B4 )
     => ( ( ord_le3146513528884898305at_nat @ A4 @ B4 )
       => ( dvd_dvd_Code_integer @ ( groups1230400874837758585nteger @ F @ A4 ) @ ( groups1230400874837758585nteger @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9057_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( dvd_dvd_Code_integer @ ( groups3455450783089532116nteger @ F @ A4 ) @ ( groups3455450783089532116nteger @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9058_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( dvd_dvd_nat @ ( groups708209901874060359at_nat @ F @ A4 ) @ ( groups708209901874060359at_nat @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9059_prod__dvd__prod__subset,axiom,
    ! [B4: set_nat,A4: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B4 )
     => ( ( ord_less_eq_set_nat @ A4 @ B4 )
       => ( dvd_dvd_int @ ( groups705719431365010083at_int @ F @ A4 ) @ ( groups705719431365010083at_int @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9060_prod__dvd__prod__subset,axiom,
    ! [B4: set_int,A4: set_int,F: int > int] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( dvd_dvd_int @ ( groups1705073143266064639nt_int @ F @ A4 ) @ ( groups1705073143266064639nt_int @ F @ B4 ) ) ) ) ).

% prod_dvd_prod_subset
thf(fact_9061_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_9062_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_9063_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_9064_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_9065_atMost__nat__numeral,axiom,
    ! [K2: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K2 ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K2 ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K2 ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9066_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_9067_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_9068_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9069_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9070_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9071_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9072_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9073_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_9074_prod_OatLeastAtMost__rev,axiom,
    ! [G: nat > int,N: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I4: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I4 ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_9075_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > complex,H: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K2 ) @ one_one_complex @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups6464643781859351333omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_9076_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > real,H: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ one_one_real @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups129246275422532515t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_9077_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ one_one_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups73079841787564623at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_9078_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups708209901874060359at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_9079_prod_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ one_one_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups705719431365010083at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_9080_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_9081_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_9082_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_9083_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_9084_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_9085_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_9086_less__1__prod2,axiom,
    ! [I5: set_o,I: $o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( member_o @ I @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F @ I ) )
         => ( ! [I3: $o] :
                ( ( member_o @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_9087_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_9088_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_9089_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_9090_sum__choose__upper,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M2 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ N ) @ ( suc @ M2 ) ) ) ).

% sum_choose_upper
thf(fact_9091_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_9092_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_9093_less__1__prod,axiom,
    ! [I5: set_o,F: $o > real] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_real @ one_one_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ one_one_real @ ( groups234877984723959775o_real @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_9094_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_9095_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_9096_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_9097_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_9098_less__1__prod,axiom,
    ! [I5: set_o,F: $o > rat] :
      ( ( finite_finite_o @ I5 )
     => ( ( I5 != bot_bot_set_o )
       => ( ! [I3: $o] :
              ( ( member_o @ I3 @ I5 )
             => ( ord_less_rat @ one_one_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ one_one_rat @ ( groups2869687844427037835_o_rat @ F @ I5 ) ) ) ) ) ).

% less_1_prod
thf(fact_9099_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_9100_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_9101_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups766887009212190081x_real @ G @ A4 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups766887009212190081x_real @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9102_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups225925009352817453ex_rat @ G @ A4 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups225925009352817453ex_rat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9103_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups861055069439313189ex_nat @ G @ A4 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9104_prod_Osubset__diff,axiom,
    ! [B4: set_complex,A4: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B4 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups858564598930262913ex_int @ G @ A4 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G @ ( minus_811609699411566653omplex @ A4 @ B4 ) ) @ ( groups858564598930262913ex_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9105_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > real] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups129246275422532515t_real @ G @ A4 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups129246275422532515t_real @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9106_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ A4 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9107_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > nat] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups708209901874060359at_nat @ G @ A4 )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups708209901874060359at_nat @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9108_prod_Osubset__diff,axiom,
    ! [B4: set_nat,A4: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B4 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups705719431365010083at_int @ G @ A4 )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( minus_minus_set_nat @ A4 @ B4 ) ) @ ( groups705719431365010083at_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9109_prod_Osubset__diff,axiom,
    ! [B4: set_int,A4: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B4 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1705073143266064639nt_int @ G @ A4 )
          = ( times_times_int @ ( groups1705073143266064639nt_int @ G @ ( minus_minus_set_int @ A4 @ B4 ) ) @ ( groups1705073143266064639nt_int @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9110_prod_Osubset__diff,axiom,
    ! [B4: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > real] :
      ( ( ord_le3146513528884898305at_nat @ B4 @ A4 )
     => ( ( finite6177210948735845034at_nat @ A4 )
       => ( ( groups6036352826371341000t_real @ G @ A4 )
          = ( times_times_real @ ( groups6036352826371341000t_real @ G @ ( minus_1356011639430497352at_nat @ A4 @ B4 ) ) @ ( groups6036352826371341000t_real @ G @ B4 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_9111_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > complex,H: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ T3 )
              = ( groups713298508707869441omplex @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9112_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups4859619685533338977omplex @ G @ T3 )
              = ( groups4859619685533338977omplex @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9113_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex,H: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ T3 )
              = ( groups7440179247065528705omplex @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9114_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ T3 )
              = ( groups3708469109370488835omplex @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9115_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ T3 )
              = ( groups1681761925125756287l_real @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9116_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups234877984723959775o_real @ G @ T3 )
              = ( groups234877984723959775o_real @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9117_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ T3 )
              = ( groups2316167850115554303t_real @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9118_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ T3 )
              = ( groups766887009212190081x_real @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9119_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ T3 )
              = ( groups4061424788464935467al_rat @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9120_prod_Omono__neutral__cong__right,axiom,
    ! [T3: set_o,S3: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ T3 )
              = ( groups2869687844427037835_o_rat @ H @ S3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_right
thf(fact_9121_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_complex ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups713298508707869441omplex @ G @ S3 )
              = ( groups713298508707869441omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9122_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > complex,G: $o > complex] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_complex ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups4859619685533338977omplex @ G @ S3 )
              = ( groups4859619685533338977omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9123_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H: int > complex,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_complex ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups7440179247065528705omplex @ G @ S3 )
              = ( groups7440179247065528705omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9124_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: complex > complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_complex ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups3708469109370488835omplex @ G @ S3 )
              = ( groups3708469109370488835omplex @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9125_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_real ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups1681761925125756287l_real @ G @ S3 )
              = ( groups1681761925125756287l_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9126_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > real,G: $o > real] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_real ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups234877984723959775o_real @ G @ S3 )
              = ( groups234877984723959775o_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9127_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_int,S3: set_int,H: int > real,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X5: int] :
              ( ( member_int @ X5 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_real ) )
         => ( ! [X5: int] :
                ( ( member_int @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups2316167850115554303t_real @ G @ S3 )
              = ( groups2316167850115554303t_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9128_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_real ) )
         => ( ! [X5: complex] :
                ( ( member_complex @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups766887009212190081x_real @ G @ S3 )
              = ( groups766887009212190081x_real @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9129_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X5: real] :
              ( ( member_real @ X5 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_rat ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups4061424788464935467al_rat @ G @ S3 )
              = ( groups4061424788464935467al_rat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9130_prod_Omono__neutral__cong__left,axiom,
    ! [T3: set_o,S3: set_o,H: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ T3 )
     => ( ( ord_less_eq_set_o @ S3 @ T3 )
       => ( ! [X5: $o] :
              ( ( member_o @ X5 @ ( minus_minus_set_o @ T3 @ S3 ) )
             => ( ( H @ X5 )
                = one_one_rat ) )
         => ( ! [X5: $o] :
                ( ( member_o @ X5 @ S3 )
               => ( ( G @ X5 )
                  = ( H @ X5 ) ) )
           => ( ( groups2869687844427037835_o_rat @ G @ S3 )
              = ( groups2869687844427037835_o_rat @ H @ T3 ) ) ) ) ) ) ).

% prod.mono_neutral_cong_left
thf(fact_9131_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ T3 )
            = ( groups3708469109370488835omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9132_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ T3 )
            = ( groups766887009212190081x_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9133_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ T3 )
            = ( groups225925009352817453ex_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9134_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ T3 )
            = ( groups861055069439313189ex_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9135_prod_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ T3 )
            = ( groups858564598930262913ex_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9136_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ T3 )
            = ( groups6464643781859351333omplex @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9137_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ T3 )
            = ( groups129246275422532515t_real @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9138_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ T3 )
            = ( groups73079841787564623at_rat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9139_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ T3 )
            = ( groups708209901874060359at_nat @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9140_prod_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ T3 )
            = ( groups705719431365010083at_int @ G @ S3 ) ) ) ) ) ).

% prod.mono_neutral_right
thf(fact_9141_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G @ S3 )
            = ( groups3708469109370488835omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9142_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G @ S3 )
            = ( groups766887009212190081x_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9143_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups225925009352817453ex_rat @ G @ S3 )
            = ( groups225925009352817453ex_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9144_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups861055069439313189ex_nat @ G @ S3 )
            = ( groups861055069439313189ex_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9145_prod_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X5: complex] :
              ( ( member_complex @ X5 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups858564598930262913ex_int @ G @ S3 )
            = ( groups858564598930262913ex_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9146_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G @ S3 )
            = ( groups6464643781859351333omplex @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9147_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G @ S3 )
            = ( groups129246275422532515t_real @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9148_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_rat ) )
         => ( ( groups73079841787564623at_rat @ G @ S3 )
            = ( groups73079841787564623at_rat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9149_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_nat ) )
         => ( ( groups708209901874060359at_nat @ G @ S3 )
            = ( groups708209901874060359at_nat @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9150_prod_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X5: nat] :
              ( ( member_nat @ X5 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X5 )
                = one_one_int ) )
         => ( ( groups705719431365010083at_int @ G @ S3 )
            = ( groups705719431365010083at_int @ G @ T3 ) ) ) ) ) ).

% prod.mono_neutral_left
thf(fact_9151_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > complex,H: real > complex] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ C2 )
                  = ( groups713298508707869441omplex @ H @ C2 ) )
               => ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9152_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups4859619685533338977omplex @ G @ C2 )
                  = ( groups4859619685533338977omplex @ H @ C2 ) )
               => ( ( groups4859619685533338977omplex @ G @ A4 )
                  = ( groups4859619685533338977omplex @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9153_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > complex,H: int > complex] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ C2 )
                  = ( groups7440179247065528705omplex @ H @ C2 ) )
               => ( ( groups7440179247065528705omplex @ G @ A4 )
                  = ( groups7440179247065528705omplex @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9154_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ C2 )
                  = ( groups3708469109370488835omplex @ H @ C2 ) )
               => ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9155_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ C2 )
                  = ( groups1681761925125756287l_real @ H @ C2 ) )
               => ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9156_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups234877984723959775o_real @ G @ C2 )
                  = ( groups234877984723959775o_real @ H @ C2 ) )
               => ( ( groups234877984723959775o_real @ G @ A4 )
                  = ( groups234877984723959775o_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9157_prod_Osame__carrierI,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ C2 )
                  = ( groups2316167850115554303t_real @ H @ C2 ) )
               => ( ( groups2316167850115554303t_real @ G @ A4 )
                  = ( groups2316167850115554303t_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9158_prod_Osame__carrierI,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ C2 )
                  = ( groups766887009212190081x_real @ H @ C2 ) )
               => ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9159_prod_Osame__carrierI,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ C2 )
                  = ( groups4061424788464935467al_rat @ H @ C2 ) )
               => ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9160_prod_Osame__carrierI,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ C2 )
                  = ( groups2869687844427037835_o_rat @ H @ C2 ) )
               => ( ( groups2869687844427037835_o_rat @ G @ A4 )
                  = ( groups2869687844427037835_o_rat @ H @ B4 ) ) ) ) ) ) ) ) ).

% prod.same_carrierI
thf(fact_9161_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > complex,H: real > complex] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups713298508707869441omplex @ G @ A4 )
                  = ( groups713298508707869441omplex @ H @ B4 ) )
                = ( ( groups713298508707869441omplex @ G @ C2 )
                  = ( groups713298508707869441omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9162_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > complex,H: $o > complex] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups4859619685533338977omplex @ G @ A4 )
                  = ( groups4859619685533338977omplex @ H @ B4 ) )
                = ( ( groups4859619685533338977omplex @ G @ C2 )
                  = ( groups4859619685533338977omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9163_prod_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > complex,H: int > complex] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups7440179247065528705omplex @ G @ A4 )
                  = ( groups7440179247065528705omplex @ H @ B4 ) )
                = ( ( groups7440179247065528705omplex @ G @ C2 )
                  = ( groups7440179247065528705omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9164_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_complex ) )
             => ( ( ( groups3708469109370488835omplex @ G @ A4 )
                  = ( groups3708469109370488835omplex @ H @ B4 ) )
                = ( ( groups3708469109370488835omplex @ G @ C2 )
                  = ( groups3708469109370488835omplex @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9165_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > real,H: real > real] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups1681761925125756287l_real @ G @ A4 )
                  = ( groups1681761925125756287l_real @ H @ B4 ) )
                = ( ( groups1681761925125756287l_real @ G @ C2 )
                  = ( groups1681761925125756287l_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9166_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > real,H: $o > real] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups234877984723959775o_real @ G @ A4 )
                  = ( groups234877984723959775o_real @ H @ B4 ) )
                = ( ( groups234877984723959775o_real @ G @ C2 )
                  = ( groups234877984723959775o_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9167_prod_Osame__carrier,axiom,
    ! [C2: set_int,A4: set_int,B4: set_int,G: int > real,H: int > real] :
      ( ( finite_finite_int @ C2 )
     => ( ( ord_less_eq_set_int @ A4 @ C2 )
       => ( ( ord_less_eq_set_int @ B4 @ C2 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups2316167850115554303t_real @ G @ A4 )
                  = ( groups2316167850115554303t_real @ H @ B4 ) )
                = ( ( groups2316167850115554303t_real @ G @ C2 )
                  = ( groups2316167850115554303t_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9168_prod_Osame__carrier,axiom,
    ! [C2: set_complex,A4: set_complex,B4: set_complex,G: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C2 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C2 )
       => ( ( ord_le211207098394363844omplex @ B4 @ C2 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_real ) )
             => ( ( ( groups766887009212190081x_real @ G @ A4 )
                  = ( groups766887009212190081x_real @ H @ B4 ) )
                = ( ( groups766887009212190081x_real @ G @ C2 )
                  = ( groups766887009212190081x_real @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9169_prod_Osame__carrier,axiom,
    ! [C2: set_real,A4: set_real,B4: set_real,G: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C2 )
     => ( ( ord_less_eq_set_real @ A4 @ C2 )
       => ( ( ord_less_eq_set_real @ B4 @ C2 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups4061424788464935467al_rat @ G @ A4 )
                  = ( groups4061424788464935467al_rat @ H @ B4 ) )
                = ( ( groups4061424788464935467al_rat @ G @ C2 )
                  = ( groups4061424788464935467al_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9170_prod_Osame__carrier,axiom,
    ! [C2: set_o,A4: set_o,B4: set_o,G: $o > rat,H: $o > rat] :
      ( ( finite_finite_o @ C2 )
     => ( ( ord_less_eq_set_o @ A4 @ C2 )
       => ( ( ord_less_eq_set_o @ B4 @ C2 )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ ( minus_minus_set_o @ C2 @ A4 ) )
               => ( ( G @ A3 )
                  = one_one_rat ) )
           => ( ! [B3: $o] :
                  ( ( member_o @ B3 @ ( minus_minus_set_o @ C2 @ B4 ) )
                 => ( ( H @ B3 )
                    = one_one_rat ) )
             => ( ( ( groups2869687844427037835_o_rat @ G @ A4 )
                  = ( groups2869687844427037835_o_rat @ H @ B4 ) )
                = ( ( groups2869687844427037835_o_rat @ G @ C2 )
                  = ( groups2869687844427037835_o_rat @ H @ C2 ) ) ) ) ) ) ) ) ).

% prod.same_carrier
thf(fact_9171_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_9172_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_9173_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_9174_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_9175_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_9176_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_9177_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_9178_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_9179_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( G @ M2 ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_9180_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G @ M2 ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_9181_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G @ M2 ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_9182_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( G @ M2 ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_9183_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_9184_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_9185_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_9186_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_9187_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_real @ ( G @ M2 )
          @ ( groups129246275422532515t_real
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_9188_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_9189_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_9190_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
        = ( times_times_int @ ( G @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I4: nat] : ( G @ ( suc @ I4 ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_9191_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_9192_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_9193_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_9194_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_9195_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_9196_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_9197_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_9198_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_9199_prod_OatLeast1__atMost__eq,axiom,
    ! [G: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_9200_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N: nat,D: nat > complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( D @ I4 ) @ ( power_power_complex @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = ( D @ I4 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_9201_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N: nat,D: nat > real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( D @ I4 ) @ ( power_power_real @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_eq_nat @ I4 @ N )
           => ( ( C @ I4 )
              = ( D @ I4 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_9202_bounded__imp__summable,axiom,
    ! [A: nat > int,B4: int] :
      ( ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_9203_bounded__imp__summable,axiom,
    ! [A: nat > nat,B4: nat] :
      ( ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_9204_bounded__imp__summable,axiom,
    ! [A: nat > real,B4: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N2 ) ) @ B4 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_9205_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I4: nat] : ( groups3542108847815614940at_nat @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_9206_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( groups6591440286371151544t_real @ ( A @ I4 ) @ ( set_ord_lessThan_nat @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( A @ I4 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_9207_ivl__disj__un__one_I4_J,axiom,
    ! [L: rat,U: rat] :
      ( ( ord_less_eq_rat @ L @ U )
     => ( ( sup_sup_set_rat @ ( set_ord_lessThan_rat @ L ) @ ( set_or633870826150836451st_rat @ L @ U ) )
        = ( set_ord_atMost_rat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_9208_ivl__disj__un__one_I4_J,axiom,
    ! [L: num,U: num] :
      ( ( ord_less_eq_num @ L @ U )
     => ( ( sup_sup_set_num @ ( set_ord_lessThan_num @ L ) @ ( set_or7049704709247886629st_num @ L @ U ) )
        = ( set_ord_atMost_num @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_9209_ivl__disj__un__one_I4_J,axiom,
    ! [L: nat,U: nat] :
      ( ( ord_less_eq_nat @ L @ U )
     => ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ L ) @ ( set_or1269000886237332187st_nat @ L @ U ) )
        = ( set_ord_atMost_nat @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_9210_ivl__disj__un__one_I4_J,axiom,
    ! [L: int,U: int] :
      ( ( ord_less_eq_int @ L @ U )
     => ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ L ) @ ( set_or1266510415728281911st_int @ L @ U ) )
        = ( set_ord_atMost_int @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_9211_ivl__disj__un__one_I4_J,axiom,
    ! [L: real,U: real] :
      ( ( ord_less_eq_real @ L @ U )
     => ( ( sup_sup_set_real @ ( set_or5984915006950818249n_real @ L ) @ ( set_or1222579329274155063t_real @ L @ U ) )
        = ( set_ord_atMost_real @ U ) ) ) ).

% ivl_disj_un_one(4)
thf(fact_9212_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: $o] :
      ( ( sup_sup_set_o @ ( set_ord_lessThan_o @ U ) @ ( insert_o @ U @ bot_bot_set_o ) )
      = ( set_ord_atMost_o @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_9213_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: int] :
      ( ( sup_sup_set_int @ ( set_ord_lessThan_int @ U ) @ ( insert_int @ U @ bot_bot_set_int ) )
      = ( set_ord_atMost_int @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_9214_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: nat] :
      ( ( sup_sup_set_nat @ ( set_ord_lessThan_nat @ U ) @ ( insert_nat @ U @ bot_bot_set_nat ) )
      = ( set_ord_atMost_nat @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_9215_ivl__disj__un__singleton_I2_J,axiom,
    ! [U: real] :
      ( ( sup_sup_set_real @ ( set_or5984915006950818249n_real @ U ) @ ( insert_real @ U @ bot_bot_set_real ) )
      = ( set_ord_atMost_real @ U ) ) ).

% ivl_disj_un_singleton(2)
thf(fact_9216_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_9217_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_9218_prod__mono__strict,axiom,
    ! [A4: set_o,F: $o > real,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
              & ( ord_less_real @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_o )
         => ( ord_less_real @ ( groups234877984723959775o_real @ F @ A4 ) @ ( groups234877984723959775o_real @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_9219_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_9220_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_9221_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_9222_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_9223_prod__mono__strict,axiom,
    ! [A4: set_o,F: $o > rat,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ! [I3: $o] :
            ( ( member_o @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) )
              & ( ord_less_rat @ ( F @ I3 ) @ ( G @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_o )
         => ( ord_less_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_9224_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_9225_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_9226_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 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9227_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups8682486955453173170nteger @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9228_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > code_integer] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups1230400874837758585nteger @ F @ A4 ) )
        = ( ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9229_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups861055069439313189ex_nat @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9230_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups4077766827762148844at_nat @ F @ A4 ) )
        = ( ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9231_even__prod__iff,axiom,
    ! [A4: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups858564598930262913ex_int @ F @ A4 ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9232_even__prod__iff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > int] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups4075276357253098568at_int @ F @ A4 ) )
        = ( ? [X4: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9233_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 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9234_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 ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9235_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 ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( F @ X4 ) ) ) ) ) ) ).

% even_prod_iff
thf(fact_9236_sum__choose__lower,axiom,
    ! [R2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_9237_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > real,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9238_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > real,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups234877984723959775o_real @ G @ ( insert_o @ X3 @ A4 ) )
        = ( times_times_real @ ( G @ X3 ) @ ( groups234877984723959775o_real @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9239_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > real,X3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A4 ) )
        = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9240_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: nat > real,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A4 ) )
        = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9241_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > rat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( times_times_rat @ ( G @ X3 ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9242_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > rat,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups2869687844427037835_o_rat @ G @ ( insert_o @ X3 @ A4 ) )
        = ( times_times_rat @ ( G @ X3 ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9243_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G: int > rat,X3: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G @ ( insert_int @ X3 @ A4 ) )
        = ( times_times_rat @ ( G @ X3 ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9244_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G: nat > rat,X3: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A4 ) )
        = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9245_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G: complex > nat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups861055069439313189ex_nat @ G @ ( insert_complex @ X3 @ A4 ) )
        = ( times_times_nat @ ( G @ X3 ) @ ( groups861055069439313189ex_nat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9246_prod_Oinsert__remove,axiom,
    ! [A4: set_o,G: $o > nat,X3: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( groups3504817904513533571_o_nat @ G @ ( insert_o @ X3 @ A4 ) )
        = ( times_times_nat @ ( G @ X3 ) @ ( groups3504817904513533571_o_nat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_9247_prod_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups1681761925125756287l_real @ G @ A4 )
          = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9248_prod_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups766887009212190081x_real @ G @ A4 )
          = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9249_prod_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X3 @ A4 )
       => ( ( groups234877984723959775o_real @ G @ A4 )
          = ( times_times_real @ ( G @ X3 ) @ ( groups234877984723959775o_real @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9250_prod_Oremove,axiom,
    ! [A4: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X3 @ A4 )
       => ( ( groups2316167850115554303t_real @ G @ A4 )
          = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9251_prod_Oremove,axiom,
    ! [A4: set_nat,X3: nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( groups129246275422532515t_real @ G @ A4 )
          = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9252_prod_Oremove,axiom,
    ! [A4: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X3 @ A4 )
       => ( ( groups4061424788464935467al_rat @ G @ A4 )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ ( minus_minus_set_real @ A4 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9253_prod_Oremove,axiom,
    ! [A4: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X3 @ A4 )
       => ( ( groups225925009352817453ex_rat @ G @ A4 )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups225925009352817453ex_rat @ G @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9254_prod_Oremove,axiom,
    ! [A4: set_o,X3: $o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( member_o @ X3 @ A4 )
       => ( ( groups2869687844427037835_o_rat @ G @ A4 )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups2869687844427037835_o_rat @ G @ ( minus_minus_set_o @ A4 @ ( insert_o @ X3 @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9255_prod_Oremove,axiom,
    ! [A4: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X3 @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G @ A4 )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups1072433553688619179nt_rat @ G @ ( minus_minus_set_int @ A4 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9256_prod_Oremove,axiom,
    ! [A4: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X3 @ A4 )
       => ( ( groups73079841787564623at_rat @ G @ A4 )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_9257_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups766887009212190081x_real @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( times_times_real @ ( groups766887009212190081x_real @ G @ A4 ) @ ( groups766887009212190081x_real @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9258_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > real] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups234877984723959775o_real @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( times_times_real @ ( groups234877984723959775o_real @ G @ A4 ) @ ( groups234877984723959775o_real @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9259_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( inf_inf_set_nat @ A4 @ B4 )
            = bot_bot_set_nat )
         => ( ( groups129246275422532515t_real @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( times_times_real @ ( groups129246275422532515t_real @ G @ A4 ) @ ( groups129246275422532515t_real @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9260_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,G: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( ( inf_inf_set_int @ A4 @ B4 )
            = bot_bot_set_int )
         => ( ( groups2316167850115554303t_real @ G @ ( sup_sup_set_int @ A4 @ B4 ) )
            = ( times_times_real @ ( groups2316167850115554303t_real @ G @ A4 ) @ ( groups2316167850115554303t_real @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9261_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups225925009352817453ex_rat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( times_times_rat @ ( groups225925009352817453ex_rat @ G @ A4 ) @ ( groups225925009352817453ex_rat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9262_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > rat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups2869687844427037835_o_rat @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( times_times_rat @ ( groups2869687844427037835_o_rat @ G @ A4 ) @ ( groups2869687844427037835_o_rat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9263_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B4: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B4 )
       => ( ( ( inf_inf_set_nat @ A4 @ B4 )
            = bot_bot_set_nat )
         => ( ( groups73079841787564623at_rat @ G @ ( sup_sup_set_nat @ A4 @ B4 ) )
            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ A4 ) @ ( groups73079841787564623at_rat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9264_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B4: set_int,G: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B4 )
       => ( ( ( inf_inf_set_int @ A4 @ B4 )
            = bot_bot_set_int )
         => ( ( groups1072433553688619179nt_rat @ G @ ( sup_sup_set_int @ A4 @ B4 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G @ A4 ) @ ( groups1072433553688619179nt_rat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9265_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B4: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B4 )
       => ( ( ( inf_inf_set_complex @ A4 @ B4 )
            = bot_bot_set_complex )
         => ( ( groups861055069439313189ex_nat @ G @ ( sup_sup_set_complex @ A4 @ B4 ) )
            = ( times_times_nat @ ( groups861055069439313189ex_nat @ G @ A4 ) @ ( groups861055069439313189ex_nat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9266_prod_Ounion__disjoint,axiom,
    ! [A4: set_o,B4: set_o,G: $o > nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_finite_o @ B4 )
       => ( ( ( inf_inf_set_o @ A4 @ B4 )
            = bot_bot_set_o )
         => ( ( groups3504817904513533571_o_nat @ G @ ( sup_sup_set_o @ A4 @ B4 ) )
            = ( times_times_nat @ ( groups3504817904513533571_o_nat @ G @ A4 ) @ ( groups3504817904513533571_o_nat @ G @ B4 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_9267_choose__rising__sum_I2_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ M2 ) ) ).

% choose_rising_sum(2)
thf(fact_9268_choose__rising__sum_I1_J,axiom,
    ! [N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N @ J3 ) @ N )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N @ M2 ) @ one_one_nat ) @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9269_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_9270_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_9271_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_9272_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P2 ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P2 ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_9273_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F5: nat > nat > nat,A6: nat,B7: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B7 @ A6 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F5 @ ( plus_plus_nat @ A6 @ one_one_nat ) @ B7 @ ( F5 @ A6 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_9274_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X3: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X3 @ Xa2 @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X3 @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X3 @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_9275_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1681761925125756287l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9276_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups766887009212190081x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9277_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > real,C: $o > real] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups234877984723959775o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups234877984723959775o_real
              @ ^ [K3: $o] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups234877984723959775o_real @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9278_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups2316167850115554303t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9279_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: nat > real,C: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups129246275422532515t_real @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9280_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups4061424788464935467al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups4061424788464935467al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9281_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups225925009352817453ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9282_prod_Odelta__remove,axiom,
    ! [S3: set_o,A: $o,B: $o > rat,C: $o > rat] :
      ( ( finite_finite_o @ S3 )
     => ( ( ( member_o @ A @ S3 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) )
        & ( ~ ( member_o @ A @ S3 )
         => ( ( groups2869687844427037835_o_rat
              @ ^ [K3: $o] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups2869687844427037835_o_rat @ C @ ( minus_minus_set_o @ S3 @ ( insert_o @ A @ bot_bot_set_o ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9283_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1072433553688619179nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9284_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat,C: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups73079841787564623at_rat @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups73079841787564623at_rat @ C @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_9285_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N: nat,K2: 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 @ K2 @ N )
       => ( ( C @ K2 )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_9286_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N: nat,K2: 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 @ K2 @ N )
       => ( ( C @ K2 )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_9287_polyfun__eq__0,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X4 @ 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_9288_polyfun__eq__0,axiom,
    ! [C: nat > real,N: nat] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X4 @ 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_9289_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_9290_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_9291_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_9292_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_9293_sum__up__index__split,axiom,
    ! [F: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9294_sum__up__index__split,axiom,
    ! [F: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9295_sum__up__index__split,axiom,
    ! [F: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9296_sum__up__index__split,axiom,
    ! [F: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9297_gbinomial__parallel__sum,axiom,
    ! [A: complex,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
        @ ( 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_9298_gbinomial__parallel__sum,axiom,
    ! [A: rat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( 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_9299_gbinomial__parallel__sum,axiom,
    ! [A: real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
        @ ( 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_9300_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
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_9301_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
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.triangle_reindex_eq
thf(fact_9302_fact__eq__fact__times,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( semiri1408675320244567234ct_nat @ M2 )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N )
          @ ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9303_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > real] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9304_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > real] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups234877984723959775o_real @ F @ A4 ) @ ( groups234877984723959775o_real @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9305_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > real] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9306_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9307_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > rat] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9308_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > rat] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups2869687844427037835_o_rat @ F @ A4 ) @ ( groups2869687844427037835_o_rat @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9309_prod__mono2,axiom,
    ! [B4: set_int,A4: set_int,F: int > rat] :
      ( ( finite_finite_int @ B4 )
     => ( ( ord_less_eq_set_int @ A4 @ B4 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9310_prod__mono2,axiom,
    ! [B4: set_complex,A4: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B4 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9311_prod__mono2,axiom,
    ! [B4: set_real,A4: set_real,F: real > int] :
      ( ( finite_finite_real @ B4 )
     => ( ( ord_less_eq_set_real @ A4 @ B4 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B4 @ 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 @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9312_prod__mono2,axiom,
    ! [B4: set_o,A4: set_o,F: $o > int] :
      ( ( finite_finite_o @ B4 )
     => ( ( ord_less_eq_set_o @ A4 @ B4 )
       => ( ! [B3: $o] :
              ( ( member_o @ B3 @ ( minus_minus_set_o @ B4 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F @ B3 ) ) )
         => ( ! [A3: $o] :
                ( ( member_o @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups3502327434004483295_o_int @ F @ A4 ) @ ( groups3502327434004483295_o_int @ F @ B4 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_9313_prod__diff1,axiom,
    ! [A4: set_real,F: real > complex,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_real @ A @ A4 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A4 )
           => ( ( groups713298508707869441omplex @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups713298508707869441omplex @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9314_prod__diff1,axiom,
    ! [A4: set_complex,F: complex > complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_complex @ A @ A4 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A4 )
           => ( ( groups3708469109370488835omplex @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups3708469109370488835omplex @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9315_prod__diff1,axiom,
    ! [A4: set_o,F: $o > complex,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_o @ A @ A4 )
           => ( ( groups4859619685533338977omplex @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( divide1717551699836669952omplex @ ( groups4859619685533338977omplex @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o @ A @ A4 )
           => ( ( groups4859619685533338977omplex @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( groups4859619685533338977omplex @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9316_prod__diff1,axiom,
    ! [A4: set_int,F: int > complex,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_int @ A @ A4 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A4 )
           => ( ( groups7440179247065528705omplex @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups7440179247065528705omplex @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9317_prod__diff1,axiom,
    ! [A4: set_nat,F: nat > complex,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( F @ A )
         != zero_zero_complex )
       => ( ( ( member_nat @ A @ A4 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A4 )
           => ( ( groups6464643781859351333omplex @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups6464643781859351333omplex @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9318_prod__diff1,axiom,
    ! [A4: set_real,F: real > real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_real @ A @ A4 )
           => ( ( groups1681761925125756287l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_real @ ( groups1681761925125756287l_real @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_real @ A @ A4 )
           => ( ( groups1681761925125756287l_real @ F @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups1681761925125756287l_real @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9319_prod__diff1,axiom,
    ! [A4: set_complex,F: complex > real,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_complex @ A @ A4 )
           => ( ( groups766887009212190081x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_real @ ( groups766887009212190081x_real @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A4 )
           => ( ( groups766887009212190081x_real @ F @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups766887009212190081x_real @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9320_prod__diff1,axiom,
    ! [A4: set_o,F: $o > real,A: $o] :
      ( ( finite_finite_o @ A4 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_o @ A @ A4 )
           => ( ( groups234877984723959775o_real @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( divide_divide_real @ ( groups234877984723959775o_real @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_o @ A @ A4 )
           => ( ( groups234877984723959775o_real @ F @ ( minus_minus_set_o @ A4 @ ( insert_o @ A @ bot_bot_set_o ) ) )
              = ( groups234877984723959775o_real @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9321_prod__diff1,axiom,
    ! [A4: set_int,F: int > real,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_int @ A @ A4 )
           => ( ( groups2316167850115554303t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_real @ ( groups2316167850115554303t_real @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_int @ A @ A4 )
           => ( ( groups2316167850115554303t_real @ F @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups2316167850115554303t_real @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9322_prod__diff1,axiom,
    ! [A4: set_nat,F: nat > real,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( F @ A )
         != zero_zero_real )
       => ( ( ( member_nat @ A @ A4 )
           => ( ( groups129246275422532515t_real @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide_divide_real @ ( groups129246275422532515t_real @ F @ A4 ) @ ( F @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A4 )
           => ( ( groups129246275422532515t_real @ F @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups129246275422532515t_real @ F @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_9323_sum__choose__diagonal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N @ K3 ) @ ( minus_minus_nat @ M2 @ K3 ) )
          @ ( set_ord_atMost_nat @ M2 ) )
        = ( binomial @ ( suc @ N ) @ M2 ) ) ) ).

% sum_choose_diagonal
thf(fact_9324_vandermonde,axiom,
    ! [M2: nat,N: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M2 @ N ) @ R2 ) ) ).

% vandermonde
thf(fact_9325_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_9326_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_9327_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_9328_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_9329_polyfun__roots__finite,axiom,
    ! [C: nat > complex,K2: nat,N: nat] :
      ( ( ( C @ K2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z4 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_9330_polyfun__roots__finite,axiom,
    ! [C: nat > real,K2: nat,N: nat] :
      ( ( ( C @ K2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z4 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_9331_polyfun__finite__roots,axiom,
    ! [C: nat > complex,N: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X4 @ 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_9332_polyfun__finite__roots,axiom,
    ! [C: nat > real,N: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X4 @ 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_9333_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_9334_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_9335_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_9336_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_9337_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] :
            ~ ! [Z5: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9338_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] :
            ~ ! [Z5: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9339_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] :
            ~ ! [Z5: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_int @ ( minus_minus_int @ Z5 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9340_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] :
            ~ ! [Z5: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_real @ ( minus_minus_real @ Z5 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9341_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N: nat,A: complex] :
    ? [B3: nat > complex] :
    ! [Z5: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ Z5 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I4: nat] : ( times_times_complex @ ( B3 @ I4 ) @ ( power_power_complex @ Z5 @ 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_9342_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N: nat,A: rat] :
    ? [B3: nat > rat] :
    ! [Z5: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I4: nat] : ( times_times_rat @ ( C @ I4 ) @ ( power_power_rat @ Z5 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I4: nat] : ( times_times_rat @ ( B3 @ I4 ) @ ( power_power_rat @ Z5 @ 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_9343_polyfun__linear__factor,axiom,
    ! [C: nat > int,N: nat,A: int] :
    ? [B3: nat > int] :
    ! [Z5: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I4: nat] : ( times_times_int @ ( C @ I4 ) @ ( power_power_int @ Z5 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z5 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I4: nat] : ( times_times_int @ ( B3 @ I4 ) @ ( power_power_int @ Z5 @ 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_9344_polyfun__linear__factor,axiom,
    ! [C: nat > real,N: nat,A: real] :
    ? [B3: nat > real] :
    ! [Z5: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ Z5 @ I4 ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z5 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( times_times_real @ ( B3 @ I4 ) @ ( power_power_real @ Z5 @ 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_9345_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X3: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_complex @ ( power_power_complex @ X3 @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9346_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X3: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( power_power_rat @ X3 @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9347_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X3: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( power_power_int @ X3 @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9348_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X3: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( power_power_real @ X3 @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9349_pochhammer__prod__rev,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A6: rat,N3: nat] :
          ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( plus_plus_rat @ A6 @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N3 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_9350_pochhammer__prod__rev,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A6: real,N3: nat] :
          ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( plus_plus_real @ A6 @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N3 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_9351_pochhammer__prod__rev,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A6: nat,N3: nat] :
          ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( plus_plus_nat @ A6 @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N3 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_9352_pochhammer__prod__rev,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A6: int,N3: nat] :
          ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( plus_plus_int @ A6 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N3 @ I4 ) ) )
          @ ( set_or1269000886237332187st_nat @ one_one_nat @ N3 ) ) ) ) ).

% pochhammer_prod_rev
thf(fact_9353_fact__div__fact,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) ) ).

% fact_div_fact
thf(fact_9354_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
        @ ^ [K3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_9355_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
        @ ^ [K3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I4: nat] : ( G @ I4 @ ( minus_minus_nat @ K3 @ I4 ) )
            @ ( set_ord_atMost_nat @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.triangle_reindex
thf(fact_9356_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_9357_binomial,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial
thf(fact_9358_atLeast1__atMost__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_atMost_eq_remove0
thf(fact_9359_prod_Oin__pairs,axiom,
    ! [G: nat > real,M2: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_9360_prod_Oin__pairs,axiom,
    ! [G: nat > rat,M2: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_9361_prod_Oin__pairs,axiom,
    ! [G: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_9362_prod_Oin__pairs,axiom,
    ! [G: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_9363_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A6: nat] : ( plus_plus_rat @ ( F @ A6 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_9364_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A6: nat] : ( plus_plus_int @ ( F @ A6 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_9365_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A6: nat] : ( plus_plus_nat @ ( F @ A6 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_9366_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A6: nat] : ( plus_plus_real @ ( F @ A6 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_9367_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_9368_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_9369_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_9370_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_9371_polynomial__product,axiom,
    ! [M2: nat,A: nat > complex,N: nat,B: nat > complex,X3: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( A @ I4 ) @ ( power_power_complex @ X3 @ I4 ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R5: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_complex @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_9372_polynomial__product,axiom,
    ! [M2: nat,A: nat > rat,N: nat,B: nat > rat,X3: rat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ 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 @ M2 ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R5: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_rat @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_9373_polynomial__product,axiom,
    ! [M2: nat,A: nat > int,N: nat,B: nat > int,X3: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ 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 @ M2 ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R5: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_int @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_9374_polynomial__product,axiom,
    ! [M2: nat,A: nat > real,N: nat,B: nat > real,X3: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ 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 @ M2 ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R5: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_real @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_9375_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_9376_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_9377_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_9378_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_9379_polyfun__eq__const,axiom,
    ! [C: nat > complex,N: nat,K2: complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I4: nat] : ( times_times_complex @ ( C @ I4 ) @ ( power_power_complex @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X4 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_9380_polyfun__eq__const,axiom,
    ! [C: nat > real,N: nat,K2: real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I4: nat] : ( times_times_real @ ( C @ I4 ) @ ( power_power_real @ X4 @ I4 ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C @ X4 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_9381_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_9382_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_9383_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M2 ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M2 ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_9384_binomial__ring,axiom,
    ! [A: complex,B: complex,N: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_9385_binomial__ring,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_9386_binomial__ring,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_9387_binomial__ring,axiom,
    ! [A: nat,B: nat,N: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_9388_binomial__ring,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% binomial_ring
thf(fact_9389_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_9390_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_9391_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% pochhammer_binomial_sum
thf(fact_9392_polynomial__product__nat,axiom,
    ! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X3: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N @ J2 )
           => ( ( B @ 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 @ M2 ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9393_choose__square__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ N ) ) ).

% choose_square_sum
thf(fact_9394_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K2 ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I4: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_9395_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I4: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_9396_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K2 ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I4: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_9397_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K2 ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I4: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_9398_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K2 ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I4: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I4 ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_9399_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K2 ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_9400_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K2 ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_9401_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K2 ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_9402_sum_Ozero__middle,axiom,
    ! [P2: nat,K2: nat,G: nat > real,H: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K2 @ P2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K2 ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K2 ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_9403_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: complex,X3: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X3 @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X3 ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_9404_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: rat,X3: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K3 ) @ ( power_power_rat @ X3 @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X3 ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_9405_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: real,X3: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K3 ) @ ( power_power_real @ X3 @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X3 ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_9406_root__polyfun,axiom,
    ! [N: nat,Z2: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_int @ Z2 @ 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 @ Z2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_9407_root__polyfun,axiom,
    ! [N: nat,Z2: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_complex @ Z2 @ 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 @ Z2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_9408_root__polyfun,axiom,
    ! [N: nat,Z2: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_8256067586552552935nteger @ Z2 @ 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 @ Z2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_9409_root__polyfun,axiom,
    ! [N: nat,Z2: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_rat @ Z2 @ 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 @ Z2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_9410_root__polyfun,axiom,
    ! [N: nat,Z2: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_real @ Z2 @ 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 @ Z2 @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_9411_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_9412_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_9413_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_9414_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_9415_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_9416_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_9417_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_9418_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_9419_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_9420_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( divide_divide_rat @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_9421_gbinomial__sum__nat__pow2,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M2 ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_9422_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: complex,X3: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X3 @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X3 @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_9423_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: rat,X3: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M2 ) @ A ) @ K3 ) @ ( power_power_rat @ X3 @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X3 @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_9424_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: real,X3: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ A ) @ K3 ) @ ( power_power_real @ X3 @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X3 @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_9425_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > complex,X3: complex,Y: 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 @ Y @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9426_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > rat,X3: rat,Y: 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 @ Y @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9427_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > int,X3: int,Y: 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 @ Y @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X3 @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9428_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > real,X3: real,Y: 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 @ Y @ I4 ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X3 @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X3 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9429_binomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% binomial_r_part_sum
thf(fact_9430_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_9431_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_9432_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_9433_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_9434_Maclaurin__sin__bound,axiom,
    ! [X3: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X3 )
          @ ( groups6591440286371151544t_real
            @ ^ [M5: nat] : ( times_times_real @ ( sin_coeff @ M5 ) @ ( power_power_real @ X3 @ M5 ) )
            @ ( 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_9435_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N3: nat] : N3 ) ) ).

% of_nat_id
thf(fact_9436_forall__pos__mono__1,axiom,
    ! [P: real > $o,E: real] :
      ( ! [D4: real,E2: real] :
          ( ( ord_less_real @ D4 @ E2 )
         => ( ( P @ D4 )
           => ( P @ E2 ) ) )
     => ( ! [N2: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_9437_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9438_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_9439_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_9440_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_9441_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_9442_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_9443_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_9444_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z2 )
        = one_one_real )
     => ~ ! [T5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T5 )
           => ( ( ord_less_real @ T5 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z2
               != ( complex2 @ ( cos_real @ T5 ) @ ( sin_real @ T5 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_9445_arctan__def,axiom,
    ( arctan
    = ( ^ [Y3: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X4 )
                = Y3 ) ) ) ) ) ).

% arctan_def
thf(fact_9446_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y3: real] :
          ( the_real
          @ ^ [X4: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
              & ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X4 )
                = Y3 ) ) ) ) ) ).

% arcsin_def
thf(fact_9447_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_9448_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_9449_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_9450_complex__norm,axiom,
    ! [X3: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X3 @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_9451_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X4 )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9452_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X4: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
            & ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X4 )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9453_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_9454_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_9455_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_9456_divide__int__unfold,axiom,
    ! [L: int,K2: int,N: nat,M2: nat] :
      ( ( ( ( ( sgn_sgn_int @ L )
            = zero_zero_int )
          | ( ( sgn_sgn_int @ K2 )
            = zero_zero_int )
          | ( N = zero_zero_nat ) )
       => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ( ( sgn_sgn_int @ L )
              = zero_zero_int )
            | ( ( sgn_sgn_int @ K2 )
              = zero_zero_int )
            | ( N = zero_zero_nat ) )
       => ( ( ( ( sgn_sgn_int @ K2 )
              = ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M2 @ N ) ) ) )
          & ( ( ( sgn_sgn_int @ K2 )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M2 @ N )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9457_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_9458_nat__numeral,axiom,
    ! [K2: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% nat_numeral
thf(fact_9459_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9460_nat__neg__numeral,axiom,
    ! [K2: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9461_diff__nat__numeral,axiom,
    ! [V2: num,V3: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ ( numeral_numeral_nat @ V3 ) )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).

% diff_nat_numeral
thf(fact_9462_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X3: num,N: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9463_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X3: num,N: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9464_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_9465_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_9466_nat__numeral__diff__1,axiom,
    ! [V2: num] :
      ( ( minus_minus_nat @ ( numeral_numeral_nat @ V2 ) @ one_one_nat )
      = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V2 ) @ one_one_int ) ) ) ).

% nat_numeral_diff_1
thf(fact_9467_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_9468_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_9469_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_9470_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_9471_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I4: num] : ( nat2 @ ( numeral_numeral_int @ I4 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_9472_nat__mono,axiom,
    ! [X3: int,Y: int] :
      ( ( ord_less_eq_int @ X3 @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_9473_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_9474_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_9475_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A6: nat,B7: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A6 ) @ ( semiri1314217659103216013at_int @ B7 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_9476_nat__le__eq__zle,axiom,
    ! [W: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_9477_le__nat__iff,axiom,
    ! [K2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( ord_less_eq_nat @ N @ ( nat2 @ K2 ) )
        = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K2 ) ) ) ).

% le_nat_iff
thf(fact_9478_nat__add__distrib,axiom,
    ! [Z2: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9479_Suc__as__int,axiom,
    ( suc
    = ( ^ [A6: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A6 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9480_nat__abs__triangle__ineq,axiom,
    ! [K2: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K2 @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).

% nat_abs_triangle_ineq
thf(fact_9481_nat__power__eq,axiom,
    ! [Z2: int,N: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( power_power_int @ Z2 @ N ) )
        = ( power_power_nat @ ( nat2 @ Z2 ) @ N ) ) ) ).

% nat_power_eq
thf(fact_9482_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_9483_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_9484_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9485_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9486_sgn__power__injE,axiom,
    ! [A: real,N: nat,X3: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X3 )
     => ( ( X3
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9487_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( suc @ ( nat2 @ Z2 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_9488_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9489_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_9490_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K2: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K2
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q3 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9491_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A12: int,A23: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A12 = K3 )
              & ( A23 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q4 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q4: int] :
              ( ( A12 = K3 )
              & ( A23 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q4 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q4 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9492_eucl__rel__int_Ocases,axiom,
    ! [A1: int,A22: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A1 @ A22 @ A33 )
     => ( ( ( A22 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A1 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A22 != zero_zero_int )
               => ( A1
                 != ( times_times_int @ Q2 @ A22 ) ) ) )
         => ~ ! [R3: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R3 ) )
               => ( ( ( sgn_sgn_int @ R3 )
                    = ( sgn_sgn_int @ A22 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ A22 ) )
                   => ( A1
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9493_even__nat__iff,axiom,
    ! [K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K2 ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).

% even_nat_iff
thf(fact_9494_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_9495_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_9496_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_9497_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_9498_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_9499_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_9500_divide__numeral__i,axiom,
    ! [Z2: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z2 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z2 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_9501_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_9502_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_9503_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_9504_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_9505_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_9506_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_9507_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_9508_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_9509_complex__i__not__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( numera6690914467698888265omplex @ W ) ) ).

% complex_i_not_numeral
thf(fact_9510_complex__i__not__neg__numeral,axiom,
    ! [W: num] :
      ( imaginary_unit
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_i_not_neg_numeral
thf(fact_9511_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_9512_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N3: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N3 @ ( if_nat @ ( N3 = zero_zero_nat ) @ M5 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_9513_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M5: nat,N3: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_9514_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_9515_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_9516_bij__betw__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        @ ( set_ord_lessThan_nat @ N )
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9517_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_9518_power2__csqrt,axiom,
    ! [Z2: complex] :
      ( ( power_power_complex @ ( csqrt @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z2 ) ).

% power2_csqrt
thf(fact_9519_XOR__upper,axiom,
    ! [X3: int,N: nat,Y: 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 @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% XOR_upper
thf(fact_9520_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( plus_plus_int
          @ ( zero_n2684676970156552555ol_int
            @ ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) )
             != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_9521_bij__betw__nth__root__unity,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9522_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_9523_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_9524_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_9525_real__root__Suc__0,axiom,
    ! [X3: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
      = X3 ) ).

% real_root_Suc_0
thf(fact_9526_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_9527_or__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9528_and__minus__minus__numerals,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9529_real__root__power,axiom,
    ! [N: nat,X3: real,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X3 @ K2 ) )
        = ( power_power_real @ ( root @ N @ X3 ) @ K2 ) ) ) ).

% real_root_power
thf(fact_9530_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9531_not__int__div__2,axiom,
    ! [K2: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_9532_even__not__iff__int,axiom,
    ! [K2: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K2 ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K2 ) ) ) ).

% even_not_iff_int
thf(fact_9533_root__abs__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( abs_abs_real @ ( root @ N @ ( power_power_real @ Y @ N ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9534_and__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(4)
thf(fact_9535_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_9536_or__not__numerals_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9537_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_9538_int__numeral__or__not__num__neg,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ N ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9539_int__numeral__not__or__num__neg,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9540_numeral__or__not__num__eq,axiom,
    ! [M2: num,N: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M2 @ N ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9541_real__root__decreasing,axiom,
    ! [N: nat,N5: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ one_one_real @ X3 )
         => ( ord_less_eq_real @ ( root @ N5 @ X3 ) @ ( root @ N @ X3 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9542_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_9543_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_9544_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X3 )
         => ( ( root @ N @ X3 )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9545_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_9546_odd__real__root__unique,axiom,
    ! [N: nat,Y: real,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( ( power_power_real @ Y @ N )
          = X3 )
       => ( ( root @ N @ X3 )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9547_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_9548_and__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_9549_and__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M2 ) ) ) ).

% and_not_numerals(7)
thf(fact_9550_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_9551_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_9552_or__not__numerals_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9553_real__root__increasing,axiom,
    ! [N: nat,N5: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N5 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ X3 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X3 ) @ ( root @ N5 @ X3 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9554_and__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_9555_and__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_9556_or__not__numerals_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9557_root__sgn__power,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9558_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_9559_or__not__numerals_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9560_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 )
         => ! [Y3: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
                = X3 )
             => ( P @ Y3 ) ) ) ) ) ).

% split_root
thf(fact_9561_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_9562_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_9563_and__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_9564_or__not__numerals_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9565_or__not__numerals_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9566_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_9567_Sum__Ico__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ M2 @ N ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) @ ( times_times_nat @ M2 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9568_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9569_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_9570_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9571_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9572_atLeast0__lessThan__Suc,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% atLeast0_lessThan_Suc
thf(fact_9573_bit__push__bit__iff__int,axiom,
    ! [M2: nat,K2: int,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M2 @ K2 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9574_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q3: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q3 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9575_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N5 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9576_atLeastLessThan__add__Un,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K2 ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9577_atLeastLessThanSuc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = ( insert_nat @ N @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ N ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9578_prod__Suc__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_Suc_fact
thf(fact_9579_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_9580_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N3: nat,M5: nat] : ( times_times_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_nat_def
thf(fact_9581_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N3: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% push_bit_int_def
thf(fact_9582_atLeastLessThan__nat__numeral,axiom,
    ! [M2: nat,K2: num] :
      ( ( ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
          = ( insert_nat @ ( pred_numeral @ K2 ) @ ( set_or4665077453230672383an_nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ ( pred_numeral @ K2 ) )
       => ( ( set_or4665077453230672383an_nat @ M2 @ ( numeral_numeral_nat @ K2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9583_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_9584_atLeast1__lessThan__eq__remove0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% atLeast1_lessThan_eq_remove0
thf(fact_9585_sum__power2,axiom,
    ! [K2: nat] :
      ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K2 ) )
      = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 ) @ one_one_nat ) ) ).

% sum_power2
thf(fact_9586_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: 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 @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I4: nat] : ( times_times_nat @ ( A @ I4 ) @ ( B @ I4 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9587_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X8: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M5: nat] :
          ( ( ord_less_eq_nat @ M8 @ M5 )
         => ! [N3: nat] :
              ( ( ord_less_eq_nat @ M8 @ N3 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9588_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_9589_Code__Target__Int_Opositive__def,axiom,
    code_Target_positive = numeral_numeral_int ).

% Code_Target_Int.positive_def
thf(fact_9590_csqrt_Osimps_I1_J,axiom,
    ! [Z2: complex] :
      ( ( re @ ( csqrt @ Z2 ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_9591_complex__Re__numeral,axiom,
    ! [V2: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V2 ) )
      = ( numeral_numeral_real @ V2 ) ) ).

% complex_Re_numeral
thf(fact_9592_Re__divide__numeral,axiom,
    ! [Z2: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_9593_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_9594_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_9595_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z4: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z4 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z4 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z4 ) @ ( re @ Z4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_9596_csqrt_Osimps_I2_J,axiom,
    ! [Z2: complex] :
      ( ( im @ ( csqrt @ Z2 ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z2 )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z2 ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_9597_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X4: complex,Y3: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( 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 ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y3 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( 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 ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_9598_Im__power__real,axiom,
    ! [X3: complex,N: nat] :
      ( ( ( im @ X3 )
        = zero_zero_real )
     => ( ( im @ ( power_power_complex @ X3 @ N ) )
        = zero_zero_real ) ) ).

% Im_power_real
thf(fact_9599_complex__Im__numeral,axiom,
    ! [V2: num] :
      ( ( im @ ( numera6690914467698888265omplex @ V2 ) )
      = zero_zero_real ) ).

% complex_Im_numeral
thf(fact_9600_Re__power__real,axiom,
    ! [X3: complex,N: nat] :
      ( ( ( im @ X3 )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X3 @ N ) )
        = ( power_power_real @ ( re @ X3 ) @ N ) ) ) ).

% Re_power_real
thf(fact_9601_Im__divide__numeral,axiom,
    ! [Z2: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z2 @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z2 ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_9602_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_9603_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X222: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_9604_cmod__power2,axiom,
    ! [Z2: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_9605_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_9606_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_9607_complex__eq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_9608_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z4: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_9609_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_9610_complex__neq__0,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_9611_Re__divide,axiom,
    ! [X3: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X3 @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X3 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X3 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_9612_csqrt__unique,axiom,
    ! [W: complex,Z2: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z2 )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z2 )
          = W ) ) ) ).

% csqrt_unique
thf(fact_9613_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_9614_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_9615_Im__divide,axiom,
    ! [X3: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X3 @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X3 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X3 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_9616_complex__abs__le__norm,axiom,
    ! [Z2: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z2 ) ) @ ( abs_abs_real @ ( im @ Z2 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z2 ) ) ) ).

% complex_abs_le_norm
thf(fact_9617_complex__unit__circle,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z2 ) @ ( real_V1022390504157884413omplex @ Z2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_9618_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X4: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_9619_Im__Reals__divide,axiom,
    ! [R2: complex,Z2: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R2 ) ) @ ( im @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_9620_Re__Reals__divide,axiom,
    ! [R2: complex,Z2: complex] :
      ( ( member_complex @ R2 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R2 @ Z2 ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R2 ) @ ( re @ Z2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_9621_complex__mult__cnj,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9622_complex__cnj__power,axiom,
    ! [X3: complex,N: nat] :
      ( ( cnj @ ( power_power_complex @ X3 @ N ) )
      = ( power_power_complex @ ( cnj @ X3 ) @ N ) ) ).

% complex_cnj_power
thf(fact_9623_complex__cnj__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( numera6690914467698888265omplex @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% complex_cnj_numeral
thf(fact_9624_complex__cnj__neg__numeral,axiom,
    ! [W: num] :
      ( ( cnj @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% complex_cnj_neg_numeral
thf(fact_9625_complex__mod__mult__cnj,axiom,
    ! [Z2: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9626_complex__norm__square,axiom,
    ! [Z2: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z2 @ ( cnj @ Z2 ) ) ) ).

% complex_norm_square
thf(fact_9627_complex__add__cnj,axiom,
    ! [Z2: complex] :
      ( ( plus_plus_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z2 ) ) ) ) ).

% complex_add_cnj
thf(fact_9628_complex__diff__cnj,axiom,
    ! [Z2: complex] :
      ( ( minus_minus_complex @ Z2 @ ( cnj @ Z2 ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9629_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A6: complex,B7: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A6 @ ( cnj @ B7 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B7 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9630_cnj__add__mult__eq__Re,axiom,
    ! [Z2: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z2 ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z2 @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9631_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_9632_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9633_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_9634_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_9635_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M5: num,N3: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M5 ) @ ( numera6620942414471956472nteger @ N3 ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9636_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_9637_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_9638_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9639_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9640_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9641_subset__card__intvl__is__intvl,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) )
     => ( A4
        = ( set_or4665077453230672383an_nat @ K2 @ ( plus_plus_nat @ K2 @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% subset_card_intvl_is_intvl
thf(fact_9642_less__eq__nat_Osimps_I2_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M2 ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_9643_max__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ N @ M6 ) )
        @ M2 ) ) ).

% max_Suc1
thf(fact_9644_max__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M6: nat] : ( suc @ ( ord_max_nat @ M6 @ N ) )
        @ M2 ) ) ).

% max_Suc2
thf(fact_9645_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N5: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N5 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9646_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : X4
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9647_card__nth__roots,axiom,
    ! [C: complex,N: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z4: complex] :
                  ( ( power_power_complex @ Z4 @ N )
                  = C ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_9648_card__roots__unity__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) ) )
        = N ) ) ).

% card_roots_unity_eq
thf(fact_9649_diff__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M2 @ N ) ) ) ).

% diff_Suc
thf(fact_9650_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9651_Code__Numeral_Opositive__def,axiom,
    code_positive = numera6620942414471956472nteger ).

% Code_Numeral.positive_def
thf(fact_9652_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9653_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_9654_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_9655_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S7: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S7 ) ) @ ( S7 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9656_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9657_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9658_int__of__integer__numeral,axiom,
    ! [K2: num] :
      ( ( code_int_of_integer @ ( numera6620942414471956472nteger @ K2 ) )
      = ( numeral_numeral_int @ K2 ) ) ).

% int_of_integer_numeral
thf(fact_9659_nat__of__integer__code__post_I3_J,axiom,
    ! [K2: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% nat_of_integer_code_post(3)
thf(fact_9660_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_9661_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_9662_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9663_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X23: nat] : X23 ) ) ).

% pred_def
thf(fact_9664_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S7 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S7 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9665_bezw__0,axiom,
    ! [X3: nat] :
      ( ( bezw @ X3 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9666_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9667_drop__bit__Suc__minus__bit0,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% drop_bit_Suc_minus_bit0
thf(fact_9668_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9669_drop__bit__Suc__minus__bit1,axiom,
    ! [N: nat,K2: num] :
      ( ( bit_se8568078237143864401it_int @ ( suc @ N ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K2 ) ) ) )
      = ( bit_se8568078237143864401it_int @ N @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K2 ) ) ) ) ) ).

% drop_bit_Suc_minus_bit1
thf(fact_9670_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N3: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_int_def
thf(fact_9671_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9672_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_9673_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N3: nat,M5: nat] : ( divide_divide_nat @ M5 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9674_Suc__0__mod__numeral,axiom,
    ! [K2: num] :
      ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
      = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).

% Suc_0_mod_numeral
thf(fact_9675_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M5 @ K3 ) @ ( product_Pair_nat_nat @ M5 @ ( minus_minus_nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9676_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9677_Suc__0__div__numeral,axiom,
    ! [K2: num] :
      ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K2 ) )
      = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K2 ) ) ) ).

% Suc_0_div_numeral
thf(fact_9678_Inf__nat__def1,axiom,
    ! [K5: set_nat] :
      ( ( K5 != bot_bot_set_nat )
     => ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).

% Inf_nat_def1
thf(fact_9679_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_9680_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_9681_minus__numeral__mod__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9682_numeral__mod__minus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M2 @ N ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9683_bezw__non__0,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X3 @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X3 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9684_bezw_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ 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_9685_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y3: nat] : ( if_Pro3027730157355071871nt_int @ ( Y3 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X4 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y3 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9686_bezw_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ 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_9687_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y
                  = ( 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_9688_vebt__maxt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [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 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9689_vebt__mint_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) ) ) )
           => ~ ! [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 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9690_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I2: nat] :
                  ( ( ord_less_nat @ K @ I2 )
                 => ( P @ I2 ) )
             => ( P @ K ) ) )
       => ( P @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_9691_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ( ( X3
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ $true @ Uv ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) ) )
           => ( ! [Uu: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ Uu @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) ) )
             => ( ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                     => ( ~ Y
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9692_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 ) ) )
         => ~ ! [Uw: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9693_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ! [Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ $true @ Uv ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv ) ) )
         => ( ! [Uu: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ Uu @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9694_xor__minus__numerals_I2_J,axiom,
    ! [K2: int,N: num] :
      ( ( bit_se6526347334894502574or_int @ K2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K2 @ ( neg_numeral_sub_int @ N @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9695_xor__minus__numerals_I1_J,axiom,
    ! [N: num,K2: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) @ K2 )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N @ one ) @ K2 ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9696_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_9697_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_9698_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X4 )
    @ ^ [X4: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9699_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S7: code_integer] : ( if_Pro6119634080678213985nteger @ ( S7 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S7 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9700_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
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% times_int.abs_eq
thf(fact_9701_card_Ocomp__fun__commute__on,axiom,
    ( ( comp_nat_nat_nat @ suc @ suc )
    = ( comp_nat_nat_nat @ suc @ suc ) ) ).

% card.comp_fun_commute_on
thf(fact_9702_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X5: nat,Y4: nat] :
          ( Z2
         != ( abs_Integ @ ( product_Pair_nat_nat @ X5 @ Y4 ) ) ) ).

% eq_Abs_Integ
thf(fact_9703_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9704_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N3: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9705_uminus__int_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 )
          @ X3 ) ) ) ).

% uminus_int.abs_eq
thf(fact_9706_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9707_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
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_int.abs_eq
thf(fact_9708_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
        @ ^ [X4: nat,Y3: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_eq_int.abs_eq
thf(fact_9709_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
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% plus_int.abs_eq
thf(fact_9710_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
          @ ^ [X4: nat,Y3: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% minus_int.abs_eq
thf(fact_9711_Code__Numeral_Onegative__def,axiom,
    ( code_negative
    = ( comp_C3531382070062128313er_num @ uminus1351360451143612070nteger @ numera6620942414471956472nteger ) ) ).

% Code_Numeral.negative_def
thf(fact_9712_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_9713_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M5: nat,N3: nat] :
            ( N3
            = ( suc @ M5 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9714_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_9715_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
      = Q3 ) ).

% num_of_nat_numeral_eq
thf(fact_9716_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9717_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_9718_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_9719_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_9720_num__of__nat__plus__distrib,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( num_of_nat @ ( plus_plus_nat @ M2 @ N ) )
          = ( plus_plus_num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9721_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z4: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9722_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y3: nat,Z4: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y3 @ V4 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9723_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 ) ) ) ) ).

% uminus_int_def
thf(fact_9724_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9725_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9726_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y3: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9727_Gcd__remove0__nat,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( gcd_Gcd_nat @ M7 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_9728_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9729_take__bit__numeral__minus__numeral__int,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q4: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M2 ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_int @ Q4 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M2 ) @ N ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9730_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X3: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
            @ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X3 @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X3 @ Y )
           => ( ( image_nat_nat
                @ ^ [I4: nat] : ( minus_minus_nat @ I4 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9731_take__bit__num__simps_I1_J,axiom,
    ! [M2: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M2 )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9732_bij__betw__Suc,axiom,
    ! [M7: set_nat,N5: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N5 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N5 ) ) ).

% bij_betw_Suc
thf(fact_9733_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_9734_take__bit__num__simps_I5_J,axiom,
    ! [R2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9735_image__Suc__atLeastAtMost,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9736_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9737_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_9738_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N: nat] :
      ( ( bit_take_bit_num @ N @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ one )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9739_integer__of__num__def,axiom,
    code_integer_of_num = numera6620942414471956472nteger ).

% integer_of_num_def
thf(fact_9740_image__Suc__lessThan,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ N ) ) ).

% image_Suc_lessThan
thf(fact_9741_image__Suc__atMost,axiom,
    ! [N: nat] :
      ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) )
      = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N ) ) ) ).

% image_Suc_atMost
thf(fact_9742_atLeast0__atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_atMost_Suc_eq_insert_0
thf(fact_9743_atLeast0__lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ).

% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_9744_lessThan__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% lessThan_Suc_eq_insert_0
thf(fact_9745_atMost__Suc__eq__insert__0,axiom,
    ! [N: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ N ) )
      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% atMost_Suc_eq_insert_0
thf(fact_9746_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9747_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_9748_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M5: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M5 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N3 @ ( numeral_numeral_nat @ M5 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9749_and__minus__numerals_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9750_and__minus__numerals_I7_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9751_and__minus__numerals_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9752_and__minus__numerals_I8_J,axiom,
    ! [N: num,M2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) ) @ ( numeral_numeral_int @ M2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9753_take__bit__num__simps_I4_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9754_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ N @ M2 ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9755_take__bit__num__simps_I7_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit1 @ M2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9756_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M2: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M2 ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9757_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9758_and__not__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(8)
thf(fact_9759_and__not__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% and_not_num.simps(4)
thf(fact_9760_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_9761_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_9762_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ N @ ( bit0 @ M2 ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q4: num] : ( some_num @ ( bit0 @ Q4 ) )
            @ ( bit_take_bit_num @ N3 @ M2 ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9763_and__not__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% and_not_num.simps(7)
thf(fact_9764_and__not__num__eq__Some__iff,axiom,
    ! [M2: num,N: num,Q3: num] :
      ( ( ( bit_and_not_num @ M2 @ N )
        = ( some_num @ Q3 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = ( numeral_numeral_int @ Q3 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9765_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ N @ ( bit1 @ M2 ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N3: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N3 @ M2 ) ) )
        @ N ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9766_and__not__num__eq__None__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( bit_and_not_num @ M2 @ N )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9767_int__numeral__and__not__num,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M2 ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% int_numeral_and_not_num
thf(fact_9768_int__numeral__not__and__num,axiom,
    ! [M2: num,N: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N @ M2 ) ) ) ).

% int_numeral_not_and_num
thf(fact_9769_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N3: nat,M5: num] :
          ( produc478579273971653890on_num
          @ ^ [A6: nat,X4: 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 ) ) )
                  @ X4 )
              @ A6 )
          @ ( product_Pair_nat_num @ N3 @ M5 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9770_UNIV__nat__eq,axiom,
    ( top_top_set_nat
    = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).

% UNIV_nat_eq
thf(fact_9771_and__not__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X3 = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X3 = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y != none_num ) ) )
           => ( ! [M: num] :
                  ( ( X3
                    = ( bit0 @ M ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X3
                          = ( bit1 @ M ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M @ N2 ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9772_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9773_and__not__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(5)
thf(fact_9774_and__not__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(9)
thf(fact_9775_and__not__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_and_not_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M2 @ N ) ) ) ).

% and_not_num.simps(6)
thf(fact_9776_root__def,axiom,
    ( root
    = ( ^ [N3: nat,X4: real] :
          ( if_real @ ( N3 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N3 ) )
            @ X4 ) ) ) ) ).

% root_def
thf(fact_9777_and__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X3 = one )
           => ( ? [N2: num] :
                  ( Xa2
                  = ( bit0 @ N2 ) )
             => ( Y != none_num ) ) )
         => ( ( ( X3 = one )
             => ( ? [N2: num] :
                    ( Xa2
                    = ( bit1 @ N2 ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M: num] :
                    ( X3
                    = ( bit0 @ M ) )
               => ( ( Xa2 = one )
                 => ( Y != none_num ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
                 => ( ( ? [M: num] :
                          ( X3
                          = ( bit1 @ M ) )
                     => ( ( Xa2 = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9778_UNIV__bool,axiom,
    ( top_top_set_o
    = ( insert_o @ $false @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% UNIV_bool
thf(fact_9779_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9780_and__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(5)
thf(fact_9781_and__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9782_and__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9783_and__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9784_and__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9785_and__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(6)
thf(fact_9786_and__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(8)
thf(fact_9787_and__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ).

% and_num.simps(9)
thf(fact_9788_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_9789_xor__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
        = Y )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y != none_num ) ) )
       => ( ( ( X3 = one )
           => ! [N2: num] :
                ( ( Xa2
                  = ( bit0 @ N2 ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N2 ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit1 @ N2 ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N2 ) ) ) ) )
           => ( ! [M: num] :
                  ( ( X3
                    = ( bit0 @ M ) )
                 => ( ( Xa2 = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ! [N2: num] :
                        ( ( Xa2
                          = ( bit0 @ N2 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit1 @ N2 ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X3
                          = ( bit1 @ M ) )
                       => ( ( Xa2 = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ! [N2: num] :
                              ( ( Xa2
                                = ( bit0 @ N2 ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) )
                     => ~ ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit1 @ N2 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9790_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9791_xor__num_Osimps_I5_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).

% xor_num.simps(5)
thf(fact_9792_xor__num_Osimps_I9_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ).

% xor_num.simps(9)
thf(fact_9793_xor__num_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N ) )
      = ( some_num @ ( bit1 @ N ) ) ) ).

% xor_num.simps(2)
thf(fact_9794_xor__num_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N ) )
      = ( some_num @ ( bit0 @ N ) ) ) ).

% xor_num.simps(3)
thf(fact_9795_xor__num_Osimps_I4_J,axiom,
    ! [M2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ one )
      = ( some_num @ ( bit1 @ M2 ) ) ) ).

% xor_num.simps(4)
thf(fact_9796_xor__num_Osimps_I7_J,axiom,
    ! [M2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ one )
      = ( some_num @ ( bit0 @ M2 ) ) ) ).

% xor_num.simps(7)
thf(fact_9797_xor__num_Osimps_I6_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9798_xor__num_Osimps_I8_J,axiom,
    ! [M2: num,N: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9799_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_9800_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X2 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9801_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_9802_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_9803_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_9804_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_9805_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_9806_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K2 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K2 ) ) @ ( cons_nat @ K2 @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_9807_sorted__list__of__set__atMost__Suc,axiom,
    ! [K2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K2 ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K2 ) ) @ ( cons_nat @ ( suc @ K2 ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_9808_DERIV__const__average,axiom,
    ! [A: real,B: real,V2: real > real,K2: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V2 @ K2 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( V2 @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V2 @ A ) @ ( V2 @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9809_DERIV__pow,axiom,
    ! [N: nat,X3: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X4: real] : ( power_power_real @ X4 @ 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_9810_DERIV__fun__pow,axiom,
    ! [G: real > real,M2: real,X3: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X4: real] : ( power_power_real @ ( G @ X4 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G @ X3 ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M2 )
        @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9811_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_9812_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_9813_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_9814_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_9815_DERIV__real__sqrt__generic,axiom,
    ! [X3: real,D2: real] :
      ( ( X3 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( D2
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X3 @ zero_zero_real )
           => ( D2
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9816_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_9817_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_9818_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_9819_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_9820_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9821_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T5: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
          & ( ( F @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9822_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_9823_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M: nat,T5: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T5 )
                  & ( ord_less_eq_real @ T5 @ H ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
           => ? [T5: real] :
                ( ( ord_less_real @ zero_zero_real @ T5 )
                & ( ord_less_real @ T5 @ H )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9824_Maclaurin2,axiom,
    ! [H: real,Diff: nat > real > real,F: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T5 )
                & ( ord_less_eq_real @ T5 @ H ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ? [T5: real] :
              ( ( ord_less_real @ zero_zero_real @ T5 )
              & ( ord_less_eq_real @ T5 @ H )
              & ( ( F @ H )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9825_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M: nat,T5: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ H @ T5 )
                  & ( ord_less_eq_real @ T5 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
           => ? [T5: real] :
                ( ( ord_less_real @ H @ T5 )
                & ( ord_less_real @ T5 @ zero_zero_real )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ H @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9826_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 )
         => ( ! [M: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
           => ? [T5: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T5 ) )
                & ( ord_less_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
                & ( ( F @ X3 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9827_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N: nat,X3: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M: nat,T5: real] :
            ( ( ( ord_less_nat @ M @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
       => ? [T5: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T5 ) @ ( abs_abs_real @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ X3 @ M5 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9828_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X3 )
               => ( ( ord_less_eq_real @ X3 @ B )
                 => ( ( X3 != C )
                   => ? [T5: real] :
                        ( ( ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ X3 @ T5 )
                            & ( ord_less_real @ T5 @ C ) ) )
                        & ( ~ ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ C @ T5 )
                            & ( ord_less_real @ T5 @ X3 ) ) )
                        & ( ( F @ X3 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M5 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9829_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T5: real] :
                  ( ( ord_less_real @ C @ T5 )
                  & ( ord_less_real @ T5 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9830_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M: nat,T5: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T5 )
                & ( ord_less_eq_real @ T5 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T5: real] :
                  ( ( ord_less_real @ A @ T5 )
                  & ( ord_less_real @ T5 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M5 @ C ) @ ( semiri2265585572941072030t_real @ M5 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M5 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T5 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9831_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: nat > real > real,K2: nat,B4: real] :
      ( ! [M: nat,T5: real] :
          ( ( ( ord_less_nat @ M @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T5 )
            & ( ord_less_eq_real @ T5 @ H ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T5 ) @ ( topolo2177554685111907308n_real @ T5 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K2 ) )
       => ! [M3: nat,T6: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M3 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M3 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ M3 ) ) )
                    @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N @ M3 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ M3 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M3 ) @ T6 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M3 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T6 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
                  @ ( times_times_real @ B4 @ ( divide_divide_real @ ( power_power_real @ T6 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9832_DERIV__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9833_DERIV__real__root__generic,axiom,
    ! [N: nat,X3: real,D2: 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 )
             => ( D2
                = ( 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 )
               => ( D2
                  = ( 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 )
               => ( D2
                  = ( 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 ) @ D2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9834_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X5 @ N3 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] :
                ( suminf_real
                @ ^ [N3: nat] : ( times_times_real @ ( F @ N3 ) @ ( power_power_real @ X4 @ ( suc @ N3 ) ) ) )
            @ ( suminf_real
              @ ^ [N3: nat] : ( times_times_real @ ( times_times_real @ ( F @ N3 ) @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ ( power_power_real @ X0 @ N3 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9835_list__encode_Ocases,axiom,
    ! [X3: list_nat] :
      ( ( X3 != nil_nat )
     => ~ ! [X5: nat,Xs3: list_nat] :
            ( X3
           != ( cons_nat @ X5 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_9836_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_9837_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9838_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_9839_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X4: real] : ( divide_divide_real @ ( cos_real @ X4 ) @ ( sin_real @ X4 ) )
    @ ( 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_9840_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_9841_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_9842_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_9843_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [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_9844_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_9845_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9846_nested__sequence__unique,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( G @ N2 ) )
         => ( ( filterlim_nat_real
              @ ^ [N3: nat] : ( minus_minus_real @ ( F @ N3 ) @ ( G @ N3 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [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_9847_LIMSEQ__inverse__zero,axiom,
    ! [X6: nat > real] :
      ( ! [R3: real] :
        ? [N7: nat] :
        ! [N2: nat] :
          ( ( ord_less_eq_nat @ N7 @ N2 )
         => ( ord_less_real @ R3 @ ( X6 @ N2 ) ) )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( X6 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9848_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N3: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9849_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9850_increasing__LIMSEQ,axiom,
    ! [F: nat > real,L: real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ ( F @ N2 ) @ L )
       => ( ! [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_9851_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_9852_LIMSEQ__divide__realpow__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9853_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_9854_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_9855_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N3 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9856_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9857_tendsto__exp__limit__sequentially,axiom,
    ! [X3: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ) @ N3 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X3 ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_9858_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R2: real] :
      ( filterlim_nat_real
      @ ^ [N3: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R2 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9859_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9860_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( summable_real
            @ ^ [N3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N3 ) @ ( A @ N3 ) ) ) ) ) ) ).

% summable
thf(fact_9861_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 )
     => ~ ! [K: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9862_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9863_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [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 ) ) @ N3 ) ) )
          @ ( 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_9864_zeroseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N3: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9865_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [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 ) ) @ N3 ) ) )
            @ ( 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_9866_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [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_9867_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [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
                @ ^ [N3: 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 ) ) @ N3 ) ) )
                @ ( 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
                @ ^ [N3: 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 ) ) @ N3 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9868_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N3: nat] :
              ( groups6591440286371151544t_real
              @ ^ [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 ) ) @ N3 ) @ 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_9869_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N2 ) )
       => ( ! [N2: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N2 ) ) @ ( A @ N2 ) )
         => ( filterlim_nat_real
            @ ^ [N3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [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 ) ) @ N3 ) @ 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_9870_tendsto__power__div__exp__0,axiom,
    ! [K2: nat] :
      ( filterlim_real_real
      @ ^ [X4: real] : ( divide_divide_real @ ( power_power_real @ X4 @ K2 ) @ ( exp_real @ X4 ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_9871_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_9872_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_9873_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_9874_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_top_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9875_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F: real > real,F4: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F4 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F @ X4 ) @ N )
            @ at_bot_real
            @ F4 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9876_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_9877_INT__greaterThan__UNIV,axiom,
    ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
    = bot_bot_set_nat ) ).

% INT_greaterThan_UNIV
thf(fact_9878_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9879_greaterThan__Suc,axiom,
    ! [K2: nat] :
      ( ( set_or1210151606488870762an_nat @ ( suc @ K2 ) )
      = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K2 ) @ ( insert_nat @ ( suc @ K2 ) @ bot_bot_set_nat ) ) ) ).

% greaterThan_Suc
thf(fact_9880_rat__inverse__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int,B7: int] : ( if_Pro3027730157355071871nt_int @ ( A6 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A6 ) @ B7 ) @ ( abs_abs_int @ A6 ) ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_inverse_code
thf(fact_9881_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9882_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9883_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9884_quotient__of__number_I3_J,axiom,
    ! [K2: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K2 ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9885_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_9886_normalize__denom__zero,axiom,
    ! [P2: int] :
      ( ( normalize @ ( product_Pair_int_int @ P2 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9887_quotient__of__number_I5_J,axiom,
    ! [K2: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9888_quotient__of__div,axiom,
    ! [R2: rat,N: int,D: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N @ D ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9889_rat__divide__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ D3 ) @ ( times_times_int @ C4 @ B7 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_divide_code
thf(fact_9890_rat__times__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( times_times_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A6 @ B7 ) @ ( times_times_int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_times_code
thf(fact_9891_rat__plus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A6 @ D3 ) @ ( times_times_int @ B7 @ C4 ) ) @ ( times_times_int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_plus_code
thf(fact_9892_rat__minus__code,axiom,
    ! [P2: rat,Q3: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P2 @ Q3 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int,C4: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B7: int,D3: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A6 @ D3 ) @ ( times_times_int @ B7 @ C4 ) ) @ ( times_times_int @ C4 @ D3 ) ) )
            @ ( quotient_of @ Q3 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_minus_code
thf(fact_9893_quotient__of__denom__pos,axiom,
    ! [R2: rat,P2: int,Q3: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% quotient_of_denom_pos
thf(fact_9894_rat__uminus__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A6 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_uminus_code
thf(fact_9895_rat__abs__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A6: int] : ( product_Pair_int_int @ ( abs_abs_int @ A6 ) )
        @ ( quotient_of @ P2 ) ) ) ).

% rat_abs_code
thf(fact_9896_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P2: int,Q3: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9897_normalize__crossproduct,axiom,
    ! [Q3: int,S: int,P2: int,R2: int] :
      ( ( Q3 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P2 @ S )
            = ( times_times_int @ R2 @ Q3 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9898_rat__sgn__code,axiom,
    ! [P2: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P2 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P2 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9899_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9900_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M5: nat] :
                      ( collect_nat
                      @ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M5 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9901_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( suc @ I4 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9902_eventually__sequentially__seg,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( eventually_nat
        @ ^ [N3: nat] : ( P @ ( plus_plus_nat @ N3 @ K2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_seg
thf(fact_9903_eventually__False__sequentially,axiom,
    ~ ( eventually_nat
      @ ^ [N3: nat] : $false
      @ at_top_nat ) ).

% eventually_False_sequentially
thf(fact_9904_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N6: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ N6 @ N3 )
           => ( P @ N3 ) ) ) ) ).

% eventually_sequentially
thf(fact_9905_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_9906_le__sequentially,axiom,
    ! [F4: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F4 @ at_top_nat )
      = ( ! [N6: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N6 ) @ F4 ) ) ) ).

% le_sequentially
thf(fact_9907_sequentially__offset,axiom,
    ! [P: nat > $o,K2: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I4: nat] : ( P @ ( plus_plus_nat @ I4 @ K2 ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9908_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X8: set_nat] : ( if_nat @ ( X8 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X8 ) ) ) ) ).

% Sup_nat_def
thf(fact_9909_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9910_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M5: nat,N3: nat] :
          ( if_nat @ ( N3 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N3 ) @ M5 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9911_Frct__code__post_I5_J,axiom,
    ! [K2: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K2 ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K2 ) ) ) ).

% Frct_code_post(5)
thf(fact_9912_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_9913_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_9914_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_9915_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_9916_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_9917_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9918_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9919_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_9920_Frct__code__post_I4_J,axiom,
    ! [K2: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K2 ) ) ).

% Frct_code_post(4)
thf(fact_9921_Frct__code__post_I6_J,axiom,
    ! [K2: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K2 ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K2 ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9922_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9923_gcd__Suc__0,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9924_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9925_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9926_gcd__diff2__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N @ M2 ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff2_nat
thf(fact_9927_gcd__diff1__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M2 @ N ) @ N )
        = ( gcd_gcd_nat @ M2 @ N ) ) ) ).

% gcd_diff1_nat
thf(fact_9928_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9929_Gcd__in,axiom,
    ! [A4: set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member_nat @ A3 @ A4 )
         => ( ( member_nat @ B3 @ A4 )
           => ( member_nat @ ( gcd_gcd_nat @ A3 @ B3 ) @ A4 ) ) )
     => ( ( A4 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A4 ) @ A4 ) ) ) ).

% Gcd_in
thf(fact_9930_GreatestI__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9931_Greatest__le__nat,axiom,
    ! [P: nat > $o,K2: nat,B: nat] :
      ( ( P @ K2 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9932_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y4: nat] :
            ( ( P @ Y4 )
           => ( ord_less_eq_nat @ Y4 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9933_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y4: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y4 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9934_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y4: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y4 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y4 ) @ ( times_times_nat @ B @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y4 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9935_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_9936_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_9937_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
           => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X8 )
            & ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
          @ ( 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 ) ) ) ) ) )
                   => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X8 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ Ma3 )
                    & ! [X4: nat] :
                        ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ X4 )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9938_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                  @ ( 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 ) ) ) ) ) )
                           => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                            & ! [X4: nat] :
                                ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                 => ( ( ord_less_nat @ Mi3 @ X4 )
                                    & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9939_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu: $o,Uv: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu @ Uv ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                    @ ( 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 ) ) ) ) ) )
                             => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X4: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                               => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                              & ! [X4: nat] :
                                  ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                   => ( ( ord_less_nat @ Mi3 @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9940_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                        & ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                           => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                      @ ( 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 ) ) ) ) ) )
                               => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                 => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                & ! [X4: nat] :
                                    ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                     => ( ( ord_less_nat @ Mi3 @ X4 )
                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9941_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 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9942_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu: $o,Uv: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu @ Uv ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu @ Uv ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary3 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X8 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                             => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                        @ ( 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 ) ) ) ) ) )
                                 => ( ( ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X8 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                                   => ~ ? [X8: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X8 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                                  & ! [X4: nat] :
                                      ( ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary3 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9943_gcd__nat_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X3 ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( 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_9944_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
      = ( set_or1210151606488870762an_nat @ K2 ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9945_atLeast__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
      = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K2 ) @ ( insert_nat @ K2 @ bot_bot_set_nat ) ) ) ).

% atLeast_Suc
thf(fact_9946_upto_Opelims,axiom,
    ! [X3: int,Xa2: int,Y: list_int] :
      ( ( ( upto @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y
                  = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9947_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_9948_length__upto,axiom,
    ! [I: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9949_upto__rec__numeral_I1_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9950_upto__rec__numeral_I4_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9951_upto__rec__numeral_I3_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ one_one_int ) @ ( numeral_numeral_int @ N ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M2 ) ) @ ( numeral_numeral_int @ N ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9952_upto__rec__numeral_I2_J,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M2 ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M2 ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9953_less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% less_eq
thf(fact_9954_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_9955_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y3: real] : ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9956_inj__Suc,axiom,
    ! [N5: set_nat] : ( inj_on_nat_nat @ suc @ N5 ) ).

% inj_Suc
thf(fact_9957_inj__on__diff__nat,axiom,
    ! [N5: set_nat,K2: nat] :
      ( ! [N2: nat] :
          ( ( member_nat @ N2 @ N5 )
         => ( ord_less_eq_nat @ K2 @ N2 ) )
     => ( inj_on_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ K2 )
        @ N5 ) ) ).

% inj_on_diff_nat
thf(fact_9958_inj__on__set__encode,axiom,
    inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).

% inj_on_set_encode
thf(fact_9959_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_9960_pred__nat__trancl__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% pred_nat_trancl_eq_le
thf(fact_9961_filterlim__lessThan__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_lessThan_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_lessThan_at_top
thf(fact_9962_filterlim__atMost__at__top,axiom,
    filter3212408913953519116et_nat @ set_ord_atMost_nat @ ( finite3254316476582989077op_nat @ top_top_set_nat ) @ at_top_nat ).

% filterlim_atMost_at_top
thf(fact_9963_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_9964_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9965_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9966_min__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ord_min_nat @ ( numeral_numeral_nat @ K2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).

% min_numeral_Suc
thf(fact_9967_min__Suc__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ord_min_nat @ ( suc @ N ) @ ( numeral_numeral_nat @ K2 ) )
      = ( suc @ ( ord_min_nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).

% min_Suc_numeral
thf(fact_9968_min__diff,axiom,
    ! [M2: nat,I: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I ) @ ( minus_minus_nat @ N @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I ) ) ).

% min_diff
thf(fact_9969_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9970_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_9971_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q3 ) @ ( times_times_nat @ N @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_9972_min__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ N @ M6 ) )
        @ M2 ) ) ).

% min_Suc1
thf(fact_9973_min__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M6: nat] : ( suc @ ( ord_min_nat @ M6 @ N ) )
        @ M2 ) ) ).

% min_Suc2
thf(fact_9974_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_9975_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9976_take__upt,axiom,
    ! [I: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I @ N ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M2 ) ) ) ) ).

% take_upt
thf(fact_9977_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9978_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_9979_nth__upt,axiom,
    ! [I: nat,K2: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K2 ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K2 )
        = ( plus_plus_nat @ I @ K2 ) ) ) ).

% nth_upt
thf(fact_9980_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9981_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_9982_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9983_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N3: nat,M5: nat] : ( set_nat2 @ ( upt @ N3 @ ( suc @ M5 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9984_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N3: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ ( suc @ M5 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_9985_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I4: nat] : ( plus_plus_nat @ I4 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_9986_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N3: nat,M5: nat] : ( set_nat2 @ ( upt @ ( suc @ N3 ) @ M5 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9987_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N3: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N3 ) ) ) ) ) ).

% atMost_upto
thf(fact_9988_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_9989_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_9990_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K2 ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K2 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9991_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_9992_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_9993_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_9994_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_9995_sum__list__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M2 @ N ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or4665077453230672383an_nat @ M2 @ N ) ) ) ) ).

% sum_list_upt
thf(fact_9996_card__length__sum__list__rec,axiom,
    ! [M2: nat,N5: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M2 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M2 )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N5 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M2 @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N5 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M2 )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N5 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9997_card__length__sum__list,axiom,
    ! [M2: nat,N5: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N5 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N5 @ M2 ) @ one_one_nat ) @ N5 ) ) ).

% card_length_sum_list
thf(fact_9998_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_9999_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_10000_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10001_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_10002_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( order_mono_nat_nat
        @ ^ [M5: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M5 ) @ M5 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10003_prod__encode__prod__decode__aux,axiom,
    ! [K2: nat,M2: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).

% prod_encode_prod_decode_aux
thf(fact_10004_prod__encode__eq,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_encode @ X3 )
        = ( nat_prod_encode @ Y ) )
      = ( X3 = Y ) ) ).

% prod_encode_eq
thf(fact_10005_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_10006_bij__prod__encode,axiom,
    bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).

% bij_prod_encode
thf(fact_10007_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A4 ) ).

% inj_prod_encode
thf(fact_10008_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_10009_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_10010_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M5: nat,N3: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M5 @ N3 ) ) @ M5 ) ) ) ).

% prod_encode_def
thf(fact_10011_list__encode_Oelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y )
     => ( ( ( X3 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs3: list_nat] :
              ( ( X3
                = ( cons_nat @ X5 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_10012_inj__list__encode,axiom,
    ! [A4: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A4 ) ).

% inj_list_encode
thf(fact_10013_list__encode__eq,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( ( nat_list_encode @ X3 )
        = ( nat_list_encode @ Y ) )
      = ( X3 = Y ) ) ).

% list_encode_eq
thf(fact_10014_bij__list__encode,axiom,
    bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).

% bij_list_encode
thf(fact_10015_surj__list__encode,axiom,
    ( ( image_list_nat_nat @ nat_list_encode @ top_top_set_list_nat )
    = top_top_set_nat ) ).

% surj_list_encode
thf(fact_10016_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_10017_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_10018_list__encode_Opelims,axiom,
    ! [X3: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
       => ( ( ( X3 = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X5: nat,Xs3: list_nat] :
                ( ( X3
                  = ( cons_nat @ X5 @ Xs3 ) )
               => ( ( Y
                    = ( 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_10019_and__not__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit0 @ N2 ) )
                 => ( ( Y
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N2: num] :
                    ( ( Xa2
                      = ( bit1 @ N2 ) )
                   => ( ( Y = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ( ( Xa2 = one )
                     => ( ( Y
                          = ( some_num @ ( bit0 @ M ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit0 @ N2 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X3
                          = ( bit0 @ M ) )
                       => ! [N2: num] :
                            ( ( Xa2
                              = ( bit1 @ N2 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ( ( Xa2 = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
                     => ( ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit0 @ N2 ) )
                               => ( ( Y
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                      @ ( bit_and_not_num @ M @ N2 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3
                                = ( bit1 @ M ) )
                             => ! [N2: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N2 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_10020_and__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit0 @ N2 ) )
                 => ( ( Y = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N2: num] :
                    ( ( Xa2
                      = ( bit1 @ N2 ) )
                   => ( ( Y
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ( ( Xa2 = one )
                     => ( ( Y = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit0 @ N2 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X3
                          = ( bit0 @ M ) )
                       => ! [N2: num] :
                            ( ( Xa2
                              = ( bit1 @ N2 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ( ( Xa2 = one )
                           => ( ( Y
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
                     => ( ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit0 @ N2 ) )
                               => ( ( Y
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3
                                = ( bit1 @ M ) )
                             => ! [N2: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N2 ) )
                                 => ( ( Y
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_10021_xor__num_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N2: num] :
                  ( ( Xa2
                    = ( bit0 @ N2 ) )
                 => ( ( Y
                      = ( some_num @ ( bit1 @ N2 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N2 ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [N2: num] :
                    ( ( Xa2
                      = ( bit1 @ N2 ) )
                   => ( ( Y
                        = ( some_num @ ( bit0 @ N2 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N2 ) ) ) ) ) )
             => ( ! [M: num] :
                    ( ( X3
                      = ( bit0 @ M ) )
                   => ( ( Xa2 = one )
                     => ( ( Y
                          = ( some_num @ ( bit1 @ M ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ one ) ) ) ) )
               => ( ! [M: num] :
                      ( ( X3
                        = ( bit0 @ M ) )
                     => ! [N2: num] :
                          ( ( Xa2
                            = ( bit0 @ N2 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                 => ( ! [M: num] :
                        ( ( X3
                          = ( bit0 @ M ) )
                       => ! [N2: num] :
                            ( ( Xa2
                              = ( bit1 @ N2 ) )
                           => ( ( Y
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) ) ) ) ) )
                   => ( ! [M: num] :
                          ( ( X3
                            = ( bit1 @ M ) )
                         => ( ( Xa2 = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ one ) ) ) ) )
                     => ( ! [M: num] :
                            ( ( X3
                              = ( bit1 @ M ) )
                           => ! [N2: num] :
                                ( ( Xa2
                                  = ( bit0 @ N2 ) )
                               => ( ( Y
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) ) ) ) ) )
                       => ~ ! [M: num] :
                              ( ( X3
                                = ( bit1 @ M ) )
                             => ! [N2: num] :
                                  ( ( Xa2
                                    = ( bit1 @ N2 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_10022_or__not__num__neg_Opelims,axiom,
    ! [X3: num,Xa2: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X3 @ Xa2 ) )
       => ( ( ( X3 = one )
           => ( ( Xa2 = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [M: num] :
                  ( ( Xa2
                    = ( bit0 @ M ) )
                 => ( ( Y
                      = ( bit1 @ M ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M ) ) ) ) ) )
           => ( ( ( X3 = one )
               => ! [M: num] :
                    ( ( Xa2
                      = ( bit1 @ M ) )
                   => ( ( Y
                        = ( bit1 @ M ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M ) ) ) ) ) )
             => ( ! [N2: num] :
                    ( ( X3
                      = ( bit0 @ N2 ) )
                   => ( ( Xa2 = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ one ) ) ) ) )
               => ( ! [N2: num] :
                      ( ( X3
                        = ( bit0 @ N2 ) )
                     => ! [M: num] :
                          ( ( Xa2
                            = ( bit0 @ M ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit0 @ M ) ) ) ) ) )
                 => ( ! [N2: num] :
                        ( ( X3
                          = ( bit0 @ N2 ) )
                       => ! [M: num] :
                            ( ( Xa2
                              = ( bit1 @ M ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N2 ) @ ( bit1 @ M ) ) ) ) ) )
                   => ( ! [N2: num] :
                          ( ( X3
                            = ( bit1 @ N2 ) )
                         => ( ( Xa2 = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ one ) ) ) ) )
                     => ( ! [N2: num] :
                            ( ( X3
                              = ( bit1 @ N2 ) )
                           => ! [M: num] :
                                ( ( Xa2
                                  = ( bit0 @ M ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit0 @ M ) ) ) ) ) )
                       => ~ ! [N2: num] :
                              ( ( X3
                                = ( bit1 @ N2 ) )
                             => ! [M: num] :
                                  ( ( Xa2
                                    = ( bit1 @ M ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N2 ) @ ( bit1 @ M ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_10023_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y3: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y3 @ N )
                & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_10024_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_10025_DeMoivre2,axiom,
    ! [R2: real,A: real,N: nat] :
      ( ( power_power_complex @ ( rcis @ R2 @ A ) @ N )
      = ( rcis @ ( power_power_real @ R2 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre2
thf(fact_10026_divmod__nat__code,axiom,
    ( divmod_nat
    = ( ^ [M5: nat,N3: nat] :
          ( produc8678311845419106900er_nat @ code_nat_of_integer @ code_nat_of_integer
          @ ( if_Pro6119634080678213985nteger
            @ ( ( code_integer_of_nat @ M5 )
              = zero_z3403309356797280102nteger )
            @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
            @ ( if_Pro6119634080678213985nteger
              @ ( ( code_integer_of_nat @ N3 )
                = zero_z3403309356797280102nteger )
              @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( code_integer_of_nat @ M5 ) )
              @ ( code_divmod_abs @ ( code_integer_of_nat @ M5 ) @ ( code_integer_of_nat @ N3 ) ) ) ) ) ) ) ).

% divmod_nat_code
thf(fact_10027_integer__of__nat__numeral,axiom,
    ! [N: num] :
      ( ( code_integer_of_nat @ ( numeral_numeral_nat @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% integer_of_nat_numeral
thf(fact_10028_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] :
          ! [M5: nat] :
            ( ( ord_less_eq_nat @ N6 @ M5 )
           => ! [N3: nat] :
                ( ( ord_less_eq_nat @ N6 @ N3 )
               => ( P @ ( product_Pair_nat_nat @ N3 @ M5 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_10029_le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ).

% le_enumerate
thf(fact_10030_finite__le__enumerate,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ N @ ( finite_card_nat @ S3 ) )
       => ( ord_less_eq_nat @ N @ ( infini8530281810654367211te_nat @ S3 @ N ) ) ) ) ).

% finite_le_enumerate
thf(fact_10031_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I4: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I4 @ J3 ) @ M2 ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_10032_Least__eq__0,axiom,
    ! [P: nat > $o] :
      ( ( P @ zero_zero_nat )
     => ( ( ord_Least_nat @ P )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_10033_Least__Suc2,axiom,
    ! [P: nat > $o,N: nat,Q: nat > $o,M2: nat] :
      ( ( P @ N )
     => ( ( Q @ M2 )
       => ( ~ ( P @ zero_zero_nat )
         => ( ! [K: nat] :
                ( ( P @ ( suc @ K ) )
                = ( Q @ K ) )
           => ( ( ord_Least_nat @ P )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_10034_Least__Suc,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ( ( ord_Least_nat @ P )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_10035_vimage__Suc__insert__0,axiom,
    ! [A4: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A4 ) )
      = ( vimage_nat_nat @ suc @ A4 ) ) ).

% vimage_Suc_insert_0
thf(fact_10036_finite__vimage__Suc__iff,axiom,
    ! [F4: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F4 ) )
      = ( finite_finite_nat @ F4 ) ) ).

% finite_vimage_Suc_iff
thf(fact_10037_vimage__Suc__insert__Suc,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N ) @ A4 ) )
      = ( insert_nat @ N @ ( vimage_nat_nat @ suc @ A4 ) ) ) ).

% vimage_Suc_insert_Suc
thf(fact_10038_set__decode__div__2,axiom,
    ! [X3: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).

% set_decode_div_2
thf(fact_10039_set__encode__vimage__Suc,axiom,
    ! [A4: set_nat] :
      ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A4 ) )
      = ( divide_divide_nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% set_encode_vimage_Suc
thf(fact_10040_Restr__natLeq,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_10041_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_10042_Restr__natLeq2,axiom,
    ! [N: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
          @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_10043_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_10044_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_10045_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_10046_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_10047_pow_Osimps_I2_J,axiom,
    ! [X3: num,Y: num] :
      ( ( pow @ X3 @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X3 @ Y ) ) ) ).

% pow.simps(2)
thf(fact_10048_Rep__unit__induct,axiom,
    ! [Y: $o,P: $o > $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ! [X5: product_unit] : ( P @ ( product_Rep_unit @ X5 ) )
       => ( P @ Y ) ) ) ).

% Rep_unit_induct
thf(fact_10049_Abs__unit__inject,axiom,
    ! [X3: $o,Y: $o] :
      ( ( member_o @ X3 @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
       => ( ( ( product_Abs_unit @ X3 )
            = ( product_Abs_unit @ Y ) )
          = ( X3 = Y ) ) ) ) ).

% Abs_unit_inject
thf(fact_10050_Abs__unit__inverse,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ( ( product_Rep_unit @ ( product_Abs_unit @ Y ) )
        = Y ) ) ).

% Abs_unit_inverse
thf(fact_10051_Rep__unit__inject,axiom,
    ! [X3: product_unit,Y: product_unit] :
      ( ( ( product_Rep_unit @ X3 )
        = ( product_Rep_unit @ Y ) )
      = ( X3 = Y ) ) ).

% Rep_unit_inject
thf(fact_10052_Rep__unit__inverse,axiom,
    ! [X3: product_unit] :
      ( ( product_Abs_unit @ ( product_Rep_unit @ X3 ) )
      = X3 ) ).

% Rep_unit_inverse
thf(fact_10053_Rep__unit,axiom,
    ! [X3: product_unit] : ( member_o @ ( product_Rep_unit @ X3 ) @ ( insert_o @ $true @ bot_bot_set_o ) ) ).

% Rep_unit
thf(fact_10054_Abs__unit__cases,axiom,
    ! [X3: product_unit] :
      ~ ! [Y4: $o] :
          ( ( X3
            = ( product_Abs_unit @ Y4 ) )
         => ~ ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) ) ) ).

% Abs_unit_cases
thf(fact_10055_Rep__unit__cases,axiom,
    ! [Y: $o] :
      ( ( member_o @ Y @ ( insert_o @ $true @ bot_bot_set_o ) )
     => ~ ! [X5: product_unit] :
            ( Y
            = ( ~ ( product_Rep_unit @ X5 ) ) ) ) ).

% Rep_unit_cases
thf(fact_10056_Abs__unit__induct,axiom,
    ! [P: product_unit > $o,X3: product_unit] :
      ( ! [Y4: $o] :
          ( ( member_o @ Y4 @ ( insert_o @ $true @ bot_bot_set_o ) )
         => ( P @ ( product_Abs_unit @ Y4 ) ) )
     => ( P @ X3 ) ) ).

% Abs_unit_induct
thf(fact_10057_type__definition__unit,axiom,
    type_d6188575255521822967unit_o @ product_Rep_unit @ product_Abs_unit @ ( insert_o @ $true @ bot_bot_set_o ) ).

% type_definition_unit
thf(fact_10058_zero__rat__def,axiom,
    ( zero_zero_rat
    = ( abs_Rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ) ).

% zero_rat_def
thf(fact_10059_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_10060_one__rat__def,axiom,
    ( one_one_rat
    = ( abs_Rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ) ).

% one_rat_def
thf(fact_10061_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_10062_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_10063_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_10064_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_10065_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_10066_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_10067_inverse__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ).

% inverse_rat.rsp
thf(fact_10068_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
        @ at_top_nat )
     => ( ! [M: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M7 @ M )
           => ( ( ord_less_eq_nat @ M @ N2 )
             => ( ord_less_eq_real @ ( F @ M ) @ ( F @ N2 ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_10069_Bseq__mono__convergent,axiom,
    ! [X6: nat > real] :
      ( ( bfun_nat_real @ X6 @ at_top_nat )
     => ( ! [M: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M @ N2 )
           => ( ord_less_eq_real @ ( X6 @ M ) @ ( X6 @ N2 ) ) )
       => ( topolo7531315842566124627t_real @ X6 ) ) ) ).

% Bseq_mono_convergent
thf(fact_10070_convergent__realpow,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( topolo7531315842566124627t_real @ ( power_power_real @ X3 ) ) ) ) ).

% convergent_realpow
thf(fact_10071_uminus__rat_Orsp,axiom,
    ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ).

% uminus_rat.rsp
thf(fact_10072_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F: nat > real,M7: nat] :
      ( ( bfun_nat_real
        @ ^ [N3: nat] : ( F @ ( plus_plus_nat @ N3 @ M7 ) )
        @ at_top_nat )
     => ( ! [M: nat,N2: nat] :
            ( ( ord_less_eq_nat @ M7 @ M )
           => ( ( ord_less_eq_nat @ M @ N2 )
             => ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ M ) ) ) )
       => ( topolo7531315842566124627t_real @ F ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_10073_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).

% plus_rat.rsp
thf(fact_10074_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_10075_plus__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
      @ ^ [Y5: nat,Z: nat] : ( Y5 = Z ) )
    @ plus_plus_nat
    @ plus_plus_nat ) ).

% plus_natural.rsp
thf(fact_10076_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
    @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_10077_sub_Orsp,axiom,
    ( bNF_re8402795839162346335um_int
    @ ^ [Y5: num,Z: num] : ( Y5 = Z )
    @ ( bNF_re1822329894187522285nt_int
      @ ^ [Y5: num,Z: num] : ( Y5 = Z )
      @ ^ [Y5: int,Z: int] : ( Y5 = Z ) )
    @ ^ [M5: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) )
    @ ^ [M5: num,N3: num] : ( minus_minus_int @ ( numeral_numeral_int @ M5 ) @ ( numeral_numeral_int @ N3 ) ) ) ).

% sub.rsp
thf(fact_10078_Fract_Orsp,axiom,
    ( bNF_re157797125943740599nt_int
    @ ^ [Y5: int,Z: int] : ( Y5 = Z )
    @ ( bNF_re6250860962936578807nt_int
      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
      @ ratrel )
    @ ^ [A6: int,B7: int] : ( if_Pro3027730157355071871nt_int @ ( B7 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A6 @ B7 ) )
    @ ^ [A6: int,B7: int] : ( if_Pro3027730157355071871nt_int @ ( B7 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A6 @ B7 ) ) ) ).

% Fract.rsp
thf(fact_10079_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ).

% times_rat.rsp
thf(fact_10080_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_10081_inverse__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X4: product_prod_int_int] :
        ( if_Pro3027730157355071871nt_int
        @ ( ( product_fst_int_int @ X4 )
          = zero_zero_int )
        @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
        @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) )
    @ inverse_inverse_rat ) ).

% inverse_rat.transfer
thf(fact_10082_one__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ one_one_int @ one_one_int ) @ one_one_rat ).

% one_rat.transfer
thf(fact_10083_zero__rat_Otransfer,axiom,
    pcr_rat @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ zero_zero_rat ).

% zero_rat.transfer
thf(fact_10084_uminus__rat_Otransfer,axiom,
    ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat
    @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) )
    @ uminus_uminus_rat ) ).

% uminus_rat.transfer
thf(fact_10085_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_10086_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_10087_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_10088_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_10089_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z: nat] : ( Y5 = Z )
    @ pcr_int
    @ ^ [N3: nat] : ( product_Pair_nat_nat @ N3 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_10090_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_10091_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_10092_less__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ord_less_int ) ).

% less_int.transfer
thf(fact_10093_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_10094_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_10095_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y3 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V4 ) @ ( times_times_nat @ Y3 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_10096_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ Y3 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_10097_intrel__iff,axiom,
    ! [X3: nat,Y: nat,U: nat,V2: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ U @ V2 ) )
      = ( ( plus_plus_nat @ X3 @ V2 )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_10098_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_10099_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y3: nat] : ( product_Pair_nat_nat @ Y3 @ X4 ) ) ) ).

% uminus_int.rsp
thf(fact_10100_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_10101_intrel__def,axiom,
    ( intrel
    = ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] :
              ( ( plus_plus_nat @ X4 @ V4 )
              = ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% intrel_def
thf(fact_10102_less__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% less_int.rsp
thf(fact_10103_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z: $o] : ( Y5 = Z ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y3: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V4 ) @ ( plus_plus_nat @ U2 @ Y3 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_10104_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y3: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y3 @ V4 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_10105_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_10106_pair__less__iff1,axiom,
    ! [X3: nat,Y: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( product_Pair_nat_nat @ X3 @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z2 ) ) ).

% pair_less_iff1
thf(fact_10107_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_10108_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ X4 )
    @ ^ [X4: nat,Y3: nat] : ( ord_less_nat @ Y3 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_10109_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_10110_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_10111_wmin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_10112_wmax__insertI,axiom,
    ! [Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_10113_wmax__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X6 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X6 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_10114_wmin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_10115_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_10116_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_10117_smin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_10118_smax__insertI,axiom,
    ! [Y: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X6: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ X6 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_10119_smin__emptyI,axiom,
    ! [X6: set_Pr1261947904930325089at_nat] :
      ( ( X6 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X6 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_10120_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_10121_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_10122_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_10123_rat__number__expand_I5_J,axiom,
    ! [K2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ K2 ) )
      = ( fract @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) @ one_one_int ) ) ).

% rat_number_expand(5)
thf(fact_10124_normalize__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ( fract @ P2 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_10125_quotient__of__eq,axiom,
    ! [A: int,B: int,P2: int,Q3: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ( fract @ P2 @ Q3 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_10126_rat__number__expand_I3_J,axiom,
    ( numeral_numeral_rat
    = ( ^ [K3: num] : ( fract @ ( numeral_numeral_int @ K3 ) @ one_one_int ) ) ) ).

% rat_number_expand(3)
thf(fact_10127_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_10128_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_10129_Fract_Oabs__eq,axiom,
    ( fract
    = ( ^ [Xa3: int,X4: int] : ( abs_Rat @ ( if_Pro3027730157355071871nt_int @ ( X4 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ Xa3 @ X4 ) ) ) ) ) ).

% Fract.abs_eq
thf(fact_10130_Fract_Otransfer,axiom,
    ( bNF_re3461391660133120880nt_rat
    @ ^ [Y5: int,Z: int] : ( Y5 = Z )
    @ ( bNF_re2214769303045360666nt_rat
      @ ^ [Y5: int,Z: int] : ( Y5 = Z )
      @ pcr_rat )
    @ ^ [A6: int,B7: int] : ( if_Pro3027730157355071871nt_int @ ( B7 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ A6 @ B7 ) )
    @ fract ) ).

% Fract.transfer
thf(fact_10131_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_10132_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M5: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M5 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_10133_less__than__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ less_than )
      = ( ord_less_nat @ X3 @ Y ) ) ).

% less_than_iff
thf(fact_10134_natLeq__on__well__order__on,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_10135_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y3: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y3 @ N )
                & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y3: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y3 @ N )
              & ( ord_less_eq_nat @ X4 @ Y3 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_10136_prod__decode__triangle__add,axiom,
    ! [K2: nat,M2: nat] :
      ( ( nat_prod_decode @ ( plus_plus_nat @ ( nat_triangle @ K2 ) @ M2 ) )
      = ( nat_prod_decode_aux @ K2 @ M2 ) ) ).

% prod_decode_triangle_add
thf(fact_10137_prod__decode__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( nat_prod_decode @ X3 )
        = ( nat_prod_decode @ Y ) )
      = ( X3 = Y ) ) ).

% prod_decode_eq
thf(fact_10138_prod__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
      = N ) ).

% prod_decode_inverse
thf(fact_10139_prod__encode__inverse,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( nat_prod_decode @ ( nat_prod_encode @ X3 ) )
      = X3 ) ).

% prod_encode_inverse
thf(fact_10140_inj__prod__decode,axiom,
    ! [A4: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A4 ) ).

% inj_prod_decode
thf(fact_10141_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_10142_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).

% bij_prod_decode
thf(fact_10143_surj__prod__decode,axiom,
    ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% surj_prod_decode
thf(fact_10144_list__decode_Opinduct,axiom,
    ! [A0: nat,P: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P @ zero_zero_nat ) )
       => ( ! [N2: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) )
             => ( ! [X: nat,Y6: nat] :
                    ( ( ( product_Pair_nat_nat @ X @ Y6 )
                      = ( nat_prod_decode @ N2 ) )
                   => ( P @ Y6 ) )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_10145_list__decode_Oelims,axiom,
    ! [X3: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X3 )
        = Y )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N2: nat] :
              ( ( X3
                = ( suc @ N2 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X4: nat,Y3: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y3 ) )
                  @ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_10146_list__encode__inverse,axiom,
    ! [X3: list_nat] :
      ( ( nat_list_decode @ ( nat_list_encode @ X3 ) )
      = X3 ) ).

% list_encode_inverse
thf(fact_10147_list__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_list_encode @ ( nat_list_decode @ N ) )
      = N ) ).

% list_decode_inverse
thf(fact_10148_list__decode_Opsimps_I1_J,axiom,
    ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
   => ( ( nat_list_decode @ zero_zero_nat )
      = nil_nat ) ) ).

% list_decode.psimps(1)
thf(fact_10149_inj__list__decode,axiom,
    ! [A4: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A4 ) ).

% inj_list_decode
thf(fact_10150_list__decode__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( nat_list_decode @ X3 )
        = ( nat_list_decode @ Y ) )
      = ( X3 = Y ) ) ).

% list_decode_eq
thf(fact_10151_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_10152_list__decode_Opsimps_I2_J,axiom,
    ! [N: nat] :
      ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N ) )
     => ( ( nat_list_decode @ ( suc @ N ) )
        = ( produc2761476792215241774st_nat
          @ ^ [X4: nat,Y3: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y3 ) )
          @ ( nat_prod_decode @ N ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_10153_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).

% bij_list_decode
thf(fact_10154_surj__list__decode,axiom,
    ( ( image_nat_list_nat @ nat_list_decode @ top_top_set_nat )
    = top_top_set_list_nat ) ).

% surj_list_decode
thf(fact_10155_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( nat_list_decode @ ( suc @ N ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X4: nat,Y3: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y3 ) )
        @ ( nat_prod_decode @ N ) ) ) ).

% list_decode.simps(2)
thf(fact_10156_list__decode_Opelims,axiom,
    ! [X3: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X3 )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X3 )
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N2: nat] :
                ( ( X3
                  = ( suc @ N2 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X4: nat,Y3: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y3 ) )
                      @ ( nat_prod_decode @ N2 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_10157_compute__powr__real,axiom,
    ( powr_real2
    = ( ^ [B7: real,I4: real] :
          ( if_real @ ( ord_less_eq_real @ B7 @ zero_zero_real )
          @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ ^ [Uu3: product_unit] : ( powr_real2 @ B7 @ I4 ) )
          @ ( if_real
            @ ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ I4 ) )
              = I4 )
            @ ( if_real @ ( ord_less_eq_real @ zero_zero_real @ I4 ) @ ( power_power_real @ B7 @ ( nat2 @ ( archim6058952711729229775r_real @ I4 ) ) ) @ ( divide_divide_real @ one_one_real @ ( power_power_real @ B7 @ ( nat2 @ ( archim6058952711729229775r_real @ ( uminus_uminus_real @ I4 ) ) ) ) ) )
            @ ( abort_real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ zero_zero_literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
              @ ^ [Uu3: product_unit] : ( powr_real2 @ B7 @ I4 ) ) ) ) ) ) ).

% compute_powr_real
thf(fact_10158_inverse__rat__def,axiom,
    ( inverse_inverse_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X4: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int
          @ ( ( product_fst_int_int @ X4 )
            = zero_zero_int )
          @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
          @ ( product_Pair_int_int @ ( product_snd_int_int @ X4 ) @ ( product_fst_int_int @ X4 ) ) ) ) ) ).

% inverse_rat_def
thf(fact_10159_card__of__nat,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ top_top_set_nat ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_nat
thf(fact_10160_card__of__Field__natLeq,axiom,
    member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( bNF_Ca3793111618940312692of_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) ) @ bNF_Ca8665028551170535155natLeq ) @ bNF_We5258908940166488438at_nat ).

% card_of_Field_natLeq
thf(fact_10161_uminus__rat__def,axiom,
    ( uminus_uminus_rat
    = ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat
      @ ^ [X4: product_prod_int_int] : ( product_Pair_int_int @ ( uminus_uminus_int @ ( product_fst_int_int @ X4 ) ) @ ( product_snd_int_int @ X4 ) ) ) ) ).

% uminus_rat_def
thf(fact_10162_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y3 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% plus_rat_def
thf(fact_10163_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y3: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y3 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y3 ) ) ) ) ) ).

% times_rat_def
thf(fact_10164_ctwo__Cnotzero,axiom,
    ( ~ ( member444158400953824016od_o_o @ ( produc763777882069021527od_o_o @ bNF_Cardinal_ctwo @ bNF_Cardinal_czero_o ) @ bNF_We2654380646378065620so_o_o )
    & ( bNF_Ca8331644756375544342r_on_o @ ( field_o @ bNF_Cardinal_ctwo ) @ bNF_Cardinal_ctwo ) ) ).

% ctwo_Cnotzero
thf(fact_10165_ctwo__ordLess__natLeq,axiom,
    member4095101504841534314at_nat @ ( produc8517790099723286449at_nat @ bNF_Cardinal_ctwo @ bNF_Ca8665028551170535155natLeq ) @ bNF_We8182288985678559134_o_nat ).

% ctwo_ordLess_natLeq
thf(fact_10166_numeral__le__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% numeral_le_enat_iff
thf(fact_10167_plus__enat__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_p3455044024723400733d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% plus_enat_simps(1)
thf(fact_10168_enat__ord__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% enat_ord_simps(1)
thf(fact_10169_numeral__less__enat__iff,axiom,
    ! [M2: num,N: nat] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( extended_enat2 @ N ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ N ) ) ).

% numeral_less_enat_iff
thf(fact_10170_iadd__le__enat__iff,axiom,
    ! [X3: extended_enat,Y: extended_enat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y ) @ ( extended_enat2 @ N ) )
      = ( ? [Y8: nat,X9: nat] :
            ( ( X3
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y8 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_10171_numeral__eq__enat,axiom,
    ( numera1916890842035813515d_enat
    = ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral_nat @ K3 ) ) ) ) ).

% numeral_eq_enat
thf(fact_10172_Suc__ile__eq,axiom,
    ! [M2: nat,N: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N )
      = ( ord_le72135733267957522d_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).

% Suc_ile_eq
thf(fact_10173_elimnum,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% elimnum
thf(fact_10174_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ L ) )
      = ( vEBT_Node @ Info @ Deg
        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( map_VE8901447254227204932T_VEBT
            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
            @ TreeList ) )
        @ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide_nat @ L @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.simps(3)
thf(fact_10175_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu2: extended_enat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A @ B ) @ Uu2 )
      = ( vEBT_Leaf @ A @ B ) ) ).

% VEBT_internal.elim_dead.simps(1)
thf(fact_10176_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( vEBT_Leaf @ A3 @ B3 ) ) )
       => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y
                 != ( vEBT_Node @ Info2 @ Deg2
                    @ ( map_VE8901447254227204932T_VEBT
                      @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      @ TreeList2 )
                    @ ( vEBT_VEBT_elim_dead @ Summary3 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
               => ! [L4: nat] :
                    ( ( Xa2
                      = ( extended_enat2 @ L4 ) )
                   => ( Y
                     != ( vEBT_Node @ Info2 @ Deg2
                        @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                          @ ( map_VE8901447254227204932T_VEBT
                            @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            @ TreeList2 ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.elims
thf(fact_10177_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info @ Deg
        @ ( map_VE8901447254227204932T_VEBT
          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          @ TreeList )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_10178_elimcomplete,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).

% elimcomplete
thf(fact_10179_numeral__ne__infinity,axiom,
    ! [K2: num] :
      ( ( numera1916890842035813515d_enat @ K2 )
     != extend5688581933313929465d_enat ) ).

% numeral_ne_infinity
thf(fact_10180_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X3: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B3: $o,Uu: extended_enat] :
          ( X3
         != ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A3 @ B3 ) @ Uu ) )
     => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
            ( X3
           != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ extend5688581933313929465d_enat ) )
       => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT,L4: nat] :
              ( X3
             != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).

% VEBT_internal.elim_dead.cases
thf(fact_10181_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X3: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X3 @ Xa2 )
        = Y )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X3 @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y
                      = ( vEBT_Node @ Info2 @ Deg2
                        @ ( map_VE8901447254227204932T_VEBT
                          @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          @ TreeList2 )
                        @ ( vEBT_VEBT_elim_dead @ Summary3 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info2: option4927543243414619207at_nat,Deg2: nat,TreeList2: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) )
                 => ! [L4: nat] :
                      ( ( Xa2
                        = ( extended_enat2 @ L4 ) )
                     => ( ( Y
                          = ( vEBT_Node @ Info2 @ Deg2
                            @ ( take_VEBT_VEBT @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                              @ ( map_VE8901447254227204932T_VEBT
                                @ ^ [T2: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T2 @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                @ TreeList2 ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary3 @ ( extended_enat2 @ ( divide_divide_nat @ L4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary3 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_10182_plus__enat__def,axiom,
    ( plus_p3455044024723400733d_enat
    = ( ^ [M5: extended_enat,N3: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P5: nat] : ( extended_enat2 @ ( plus_plus_nat @ O @ P5 ) )
              @ extend5688581933313929465d_enat
              @ N3 )
          @ extend5688581933313929465d_enat
          @ M5 ) ) ) ).

% plus_enat_def
thf(fact_10183_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N3: nat] : ( extended_enat2 @ ( suc @ N3 ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_10184_eSuc__numeral,axiom,
    ! [K2: num] :
      ( ( extended_eSuc @ ( numera1916890842035813515d_enat @ K2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ K2 @ one ) ) ) ).

% eSuc_numeral
thf(fact_10185_enat__eSuc__iff,axiom,
    ! [Y: nat,X3: extended_enat] :
      ( ( ( extended_enat2 @ Y )
        = ( extended_eSuc @ X3 ) )
      = ( ? [N3: nat] :
            ( ( Y
              = ( suc @ N3 ) )
            & ( ( extended_enat2 @ N3 )
              = X3 ) ) ) ) ).

% enat_eSuc_iff
thf(fact_10186_eSuc__enat__iff,axiom,
    ! [X3: extended_enat,Y: nat] :
      ( ( ( extended_eSuc @ X3 )
        = ( extended_enat2 @ Y ) )
      = ( ? [N3: nat] :
            ( ( Y
              = ( suc @ N3 ) )
            & ( X3
              = ( extended_enat2 @ N3 ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_10187_eSuc__enat,axiom,
    ! [N: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( suc @ N ) ) ) ).

% eSuc_enat
thf(fact_10188_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_10189_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ K3 @ M5 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ R5 ) ) ) ) ) ) ).

% cauchy_def
thf(fact_10190_cauchyD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( cauchy @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ K @ M3 )
           => ! [N9: nat] :
                ( ( ord_less_eq_nat @ K @ N9 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M3 ) @ ( X6 @ N9 ) ) ) @ R2 ) ) ) ) ) ).

% cauchyD
thf(fact_10191_cauchyI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [M: nat] :
              ( ( ord_less_eq_nat @ K4 @ M )
             => ! [N2: nat] :
                  ( ( ord_less_eq_nat @ K4 @ N2 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X6 @ M ) @ ( X6 @ N2 ) ) ) @ R3 ) ) ) )
     => ( cauchy @ X6 ) ) ).

% cauchyI
thf(fact_10192_le__Real,axiom,
    ! [X6: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( cauchy @ Y7 )
       => ( ( ord_less_eq_real @ ( real2 @ X6 ) @ ( real2 @ Y7 ) )
          = ( ! [R5: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R5 )
               => ? [K3: nat] :
                  ! [N3: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N3 )
                   => ( ord_less_eq_rat @ ( X6 @ N3 ) @ ( plus_plus_rat @ ( Y7 @ N3 ) @ R5 ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_10193_cauchy__not__vanishes,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B3 )
            & ? [K: nat] :
              ! [N9: nat] :
                ( ( ord_less_eq_nat @ K @ N9 )
               => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X6 @ N9 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_10194_vanishesD,axiom,
    ! [X6: nat > rat,R2: rat] :
      ( ( vanishes @ X6 )
     => ( ( ord_less_rat @ zero_zero_rat @ R2 )
       => ? [K: nat] :
          ! [N9: nat] :
            ( ( ord_less_eq_nat @ K @ N9 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N9 ) ) @ R2 ) ) ) ) ).

% vanishesD
thf(fact_10195_vanishesI,axiom,
    ! [X6: nat > rat] :
      ( ! [R3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R3 )
         => ? [K4: nat] :
            ! [N2: nat] :
              ( ( ord_less_eq_nat @ K4 @ N2 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X6 @ N2 ) ) @ R3 ) ) )
     => ( vanishes @ X6 ) ) ).

% vanishesI
thf(fact_10196_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X8: nat > rat] :
        ! [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
         => ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R5 ) ) ) ) ) ).

% vanishes_def
thf(fact_10197_cauchy__not__vanishes__cases,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ~ ( vanishes @ X6 )
       => ? [B3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B3 )
            & ? [K: nat] :
                ( ! [N9: nat] :
                    ( ( ord_less_eq_nat @ K @ N9 )
                   => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X6 @ N9 ) ) ) )
                | ! [N9: nat] :
                    ( ( ord_less_eq_nat @ K @ N9 )
                   => ( ord_less_rat @ B3 @ ( X6 @ N9 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_10198_not__positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( ~ ( positive @ ( real2 @ X6 ) ) )
        = ( ! [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
             => ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_eq_rat @ ( X6 @ N3 ) @ R5 ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_10199_positive__Real,axiom,
    ! [X6: nat > rat] :
      ( ( cauchy @ X6 )
     => ( ( positive @ ( real2 @ X6 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ R5 @ ( X6 @ N3 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_10200_Real_Opositive_Oabs__eq,axiom,
    ! [X3: nat > rat] :
      ( ( realrel @ X3 @ X3 )
     => ( ( positive @ ( real2 @ X3 ) )
        = ( ? [R5: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R5 )
              & ? [K3: nat] :
                ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N3 )
                 => ( ord_less_rat @ R5 @ ( X3 @ N3 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_10201_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y5: $o,Z: $o] : ( Y5 = Z )
    @ ^ [X8: nat > rat] :
      ? [R5: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R5 )
        & ? [K3: nat] :
          ! [N3: nat] :
            ( ( ord_less_eq_nat @ K3 @ N3 )
           => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) )
    @ positive ) ).

% Real.positive.transfer
thf(fact_10202_Real_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X4: real] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ R5 @ ( rep_real @ X4 @ N3 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_10203_Real_Opositive__def,axiom,
    ( positive
    = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
      @ ^ [X8: nat > rat] :
        ? [R5: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R5 )
          & ? [K3: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K3 @ N3 )
             => ( ord_less_rat @ R5 @ ( X8 @ N3 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_10204_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_10205_normalize__stable,axiom,
    ! [Q3: int,P2: int] :
      ( ( ord_less_int @ zero_zero_int @ Q3 )
     => ( ( algebr932160517623751201me_int @ P2 @ Q3 )
       => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
          = ( product_Pair_int_int @ P2 @ Q3 ) ) ) ) ).

% normalize_stable

% Helper facts (35)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ( if_int @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y: int] :
      ( ( if_int @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y: nat] :
      ( ( if_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X3: num,Y: num] :
      ( ( if_num @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X3: num,Y: num] :
      ( ( if_num @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y: rat] :
      ( ( if_rat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y: rat] :
      ( ( if_rat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] :
      ( ( if_real @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y: real] :
      ( ( if_real @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ( if_complex @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y: complex] :
      ( ( if_complex @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X3: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X3: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X3: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X3 @ Y )
      = 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,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y )
      = 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,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y )
      = 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,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( sa
    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList2 @ summary2 ) ) ).

%------------------------------------------------------------------------------