%------------------------------------------------------------------------------
% File     : ITP276^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Uniqueness 00216_013627
% 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_00216_013627 [Des22]

% Status   : ContradictoryAxioms
% Rating   : 0.60 v8.2.0, 0.62 v8.1.0
% Syntax   : Number of formulae    : 11045 (5080 unt;1495 typ;   0 def)
%            Number of atoms       : 27536 (12178 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 116044 (2818   ~; 487   |;2080   &;99931   @)
%                                         (   0 <=>;10728  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  212 ( 211 usr)
%            Number of type conns  : 4874 (4874   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1287 (1284 usr;  72 con; 0-8 aty)
%            Number of variables   : 25413 (2067   ^;22644   !; 702   ?;25413   :)
% SPC      : TH0_CAX_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            from the van Emde Boas Trees session in the Archive of Formal
%            proofs - 
%            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
%            2022-02-18 15:42:08.029
%------------------------------------------------------------------------------
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% Explicit typings (1284)
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thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J,type,
    bNF_re1962705104956426057at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J,type,
    bNF_re895249473297799549at_rat: ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > $o ) > ( ( nat > rat ) > nat > rat ) > ( ( nat > rat ) > nat > rat ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001_Eo_001_Eo,type,
    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_062_I_062_It__Nat__Onat_Mt__Rat__Orat_J_M_062_It__Nat__Onat_Mt__Rat__Orat_J_J_001_062_It__Real__Oreal_Mt__Real__Oreal_J,type,
    bNF_re4695409256820837752l_real: ( ( nat > rat ) > real > $o ) > ( ( ( nat > rat ) > nat > rat ) > ( real > real ) > $o ) > ( ( nat > rat ) > ( nat > rat ) > nat > rat ) > ( real > real > real ) > $o ).

thf(sy_c_BNF__Def_Orel__fun_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal_001_062_It__Nat__Onat_Mt__Rat__Orat_J_001t__Real__Oreal,type,
    bNF_re3023117138289059399t_real: ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > real > $o ) > ( ( nat > rat ) > nat > rat ) > ( real > real ) > $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__Int__Oint_J_001_062_It__Int__Oint_Mt__Int__Oint_J,type,
    bNF_re711492959462206631nt_int: ( int > int > $o ) > ( ( int > int ) > ( int > int ) > $o ) > ( int > int > int ) > ( int > int > int ) > $o ).

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

thf(sy_c_BNF__Def_Orel__fun_001t__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__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    bNF_re5228765855967844073nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > ( ( product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int ) > $o ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > ( product_prod_int_int > product_prod_int_int > product_prod_int_int ) > $o ).

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

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

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

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

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

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

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

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

thf(sy_c_BNF__Def_Orel__fun_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Int__Oint_001t__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_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_We5258908940166488438at_nat: set_Pr4329608150637261639at_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__Code____Numeral__Ointeger,type,
    gbinom8545251970709558553nteger: code_integer > nat > code_integer ).

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_Oconcat__bit,type,
    bit_concat_bit: nat > int > int > int ).

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__Code____Numeral__Onatural,type,
    bit_se168947363167071951atural: nat > code_natural > code_natural ).

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__Nat__Onat,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

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

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit_001t__Code____Numeral__Onatural,type,
    bit_se1617098188084679374atural: nat > code_natural > code_natural ).

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit_001t__Code____Numeral__Onatural,type,
    bit_se7083795435491715335atural: nat > code_natural > code_natural ).

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

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

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

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

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

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

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

thf(sy_c_Code__Numeral_OSuc,type,
    code_Suc: code_natural > code_natural ).

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_Onat__of__integer,type,
    code_nat_of_integer: code_integer > nat ).

thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
    code_nat_of_natural: code_natural > nat ).

thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
    code_natural_of_nat: nat > code_natural ).

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

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

thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Nat__Onat_J,type,
    comple7399068483239264473et_nat: set_set_nat > set_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,
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thf(sy_c_Complex_Oimaginary__unit,type,
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thf(sy_c_Complex_Orcis,type,
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thf(sy_c_Countable_Onat__to__rat__surj,type,
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thf(sy_c_Countable_Onth__item__rel,type,
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thf(sy_c_Deriv_Odifferentiable_001t__Real__Oreal_001t__Real__Oreal,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

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

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

thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Int__Oint,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod_001t__Nat__Onat,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step_001t__Int__Oint,type,
    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,
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thf(sy_c_Extended__Nat_OeSuc,type,
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thf(sy_c_Extended__Nat_Oenat,type,
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thf(sy_c_Extended__Nat_Oenat_OAbs__enat,type,
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thf(sy_c_Extended__Nat_Oenat_Ocase__enat_001_Eo,type,
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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,
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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,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Int__Oint,type,
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thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Rat__Orat,type,
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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__Extended____Nat__Oenat,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,
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thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
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thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_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__Int__Oint,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
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thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Int_Ointrel,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
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thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Oenumerate_001_Eo,type,
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thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001_062_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001_Eo,type,
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thf(sy_c_List_Olist_OCons_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olist_OCons_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Num__Onum,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_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,
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thf(sy_c_List_Olist_OCons_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,
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thf(sy_c_List_Olist_OCons_001t__Real__Oreal,type,
    cons_real: real > list_real > list_real ).

thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_OCons_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_OCons_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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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,
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thf(sy_c_List_Olist_Oset_001_Eo,type,
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thf(sy_c_List_Olist_Oset_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_M_Eo_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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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,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
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thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Osize__list_001t__Typerep__Otyperep,type,
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thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001_Eo,type,
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thf(sy_c_List_Olist__update_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Olist__update_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist__update_001t__Int__Oint,type,
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thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
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thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olistrel_001_Eo_001t__Int__Oint,type,
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thf(sy_c_List_Omeasures_001t__Nat__Onat,type,
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thf(sy_c_List_Omeasures_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Omeasures_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Onth_001_Eo,type,
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thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Onth_001t__Num__Onum,type,
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thf(sy_c_List_Onth_001t__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,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Real__Oreal,type,
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thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct__lists_001_Eo,type,
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thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct__lists_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oreplicate_001_Eo,type,
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thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
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thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Orotate1_001_Eo,type,
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thf(sy_c_List_Orotate1_001t__Int__Oint,type,
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thf(sy_c_List_Orotate1_001t__Nat__Onat,type,
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thf(sy_c_List_Orotate1_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__Nat__Onat,type,
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thf(sy_c_List_Otake_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oupt,type,
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thf(sy_c_List_Oupto,type,
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thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
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thf(sy_c_List_Ozip_001_Eo_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ozip_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001_Eo,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__Real__Oreal,type,
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thf(sy_c_List_Ozip_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ozip_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_List_Ozip_001t__Real__Oreal_001_Eo,type,
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thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
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thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Ozip_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_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Nat_OSuc,type,
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thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
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thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Onat_Opred,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
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thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    size_s8736152011456118867at_nat: list_s1210847774152347623at_nat > nat ).

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep,type,
    size_size_typerep: typerep > 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_Oint__decode,type,
    nat_int_decode: nat > int ).

thf(sy_c_Nat__Bijection_Oint__encode,type,
    nat_int_encode: int > 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_Osum__decode,type,
    nat_sum_decode: nat > sum_sum_nat_nat ).

thf(sy_c_Nat__Bijection_Osum__encode,type,
    nat_sum_encode: sum_sum_nat_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_Onat__of__num,type,
    nat_of_num: num > nat ).

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_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__Code____Numeral__Onatural,type,
    numera5444537566228673987atural: num > code_natural ).

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_001_Eo,type,
    some_o: $o > option_o ).

thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
    some_int: int > option_int ).

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

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

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

thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

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__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_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_Olinear__order__on_001t__Nat__Onat,type,
    order_4473980167227706203on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Order__Relation_Opartial__order__on_001t__Nat__Onat,type,
    order_5251275573222108571on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Order__Relation_Opreorder__on_001t__Nat__Onat,type,
    order_4861654808422542329on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

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_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__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__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_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_It__Complex__Ocomplex_J,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Code____Numeral__Onatural,type,
    ord_le1926595141338095240atural: code_natural > code_natural > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(sy_c_Orderings_Oordering__top_001t__Nat__Onat,type,
    ordering_top_nat: ( nat > nat > $o ) > ( nat > nat > $o ) > nat > $o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Int__Oint_J,type,
    top_top_set_int: set_int ).

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,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).

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

thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Onatural,type,
    power_7079662738309270450atural: code_natural > nat > code_natural ).

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

thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
    power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).

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

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

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

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc3209952032786966637at_nat: ( nat > nat > nat ) > produc7248412053542808358at_nat > produc4471711990508489141at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Nat__Onat_J_Mt__Option__Ooption_It__Nat__Onat_J_J,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__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,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Complex__Ocomplex,type,
    produc414345526774272751omplex: $o > complex > produc648051720047351925omplex ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
    product_Pair_o_real: $o > real > product_prod_o_real ).

thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Code____Numeral__Onatural,type,
    produc3574140220909816553atural: code_natural > code_natural > produc7822875418678951345atural ).

thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    produc6639722614265839536atural: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001_Eo,type,
    produc2908979694703026321plex_o: complex > $o > produc6088675342482847199plex_o ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Complex__Ocomplex,type,
    produc101793102246108661omplex: complex > complex > produc4411394909380815293omplex ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Int__Oint,type,
    produc1367138851071493491ex_int: complex > int > produc6845221339535797307ex_int ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Nat__Onat,type,
    produc1369629321580543767ex_nat: complex > nat > produc1799700322190218207ex_nat ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__Real__Oreal,type,
    produc1746590499379883635x_real: complex > real > produc8892588492097263291x_real ).

thf(sy_c_Product__Type_OPair_001t__Complex__Ocomplex_001t__VEBT____Definitions__OVEBT,type,
    produc2757191886755552429T_VEBT: complex > vEBT_VEBT > produc7151440242714718331T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Complex__Ocomplex,type,
    produc7948753499206759283omplex: int > complex > produc5838698208256999739omplex ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
    produc801115645435158769t_real: int > real > produc679980390762269497t_real ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
    produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_It__Int__Oint_J,type,
    produc2951025481305455875st_int: list_o > list_int > produc1713839591385758857st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc7128876500814652583st_nat: list_o > list_nat > produc4203922736317485613st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc6043759678074843571T_VEBT: list_o > list_VEBT_VEBT > produc1922972420619397443T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Code____Numeral__Ointeger_J_001t__List__Olist_It__Code____Numeral__Ointeger_J,type,
    produc750622340256944499nteger: list_Code_integer > list_Code_integer > produc862207588354017979nteger ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_I_Eo_J,type,
    produc699922362453767013list_o: list_nat > list_o > produc149729814636038835list_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc8335345208264861441T_VEBT: list_nat > list_VEBT_VEBT > produc872621073311890639T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc5943733680697469783at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > produc6392793444374437607at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc7536900900485677911at_nat: list_s1210847774152347623at_nat > list_s1210847774152347623at_nat > produc424102278133772007at_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_I_Eo_J,type,
    produc2717590391345394939list_o: list_VEBT_VEBT > list_o > produc3962069817607390347list_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__Int__Oint_J,type,
    produc1392282695434103839st_int: list_VEBT_VEBT > list_int > produc7831203938951381541st_int ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5570133714943300547st_nat: list_VEBT_VEBT > list_nat > produc1097915047028332489st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc487386426758144856at_nat: nat > product_prod_nat_nat > produc7248412053542808358at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
    produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).

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,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

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,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
    product_Pair_real_o: real > $o > product_prod_real_o ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Complex__Ocomplex,type,
    produc1693001998875562995omplex: real > complex > produc6979889472282505531omplex ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
    produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).

thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

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,
    produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Complex__Ocomplex,type,
    produc5617778602380981643omplex: vEBT_VEBT > complex > produc8380087813684007313omplex ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Extended____Nat__Oenat,type,
    produc581526299967858633d_enat: vEBT_VEBT > extended_enat > produc7272778201969148633d_enat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).

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_Oapsnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__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,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Int__Oint,type,
    produc8211389475949308722nt_int: ( int > int > int ) > product_prod_int_int > int ).

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,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__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,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

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

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_Oscomp_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J_J,type,
    produc5538323210962509403atural: ( produc7822875418678951345atural > produc5835291356934675326atural ) > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > produc7822875418678951345atural > produc5835291356934675326atural ).

thf(sy_c_Random_Oiterate_001t__Code____Numeral__Onatural_001t__Product____Type__Oprod_It__Code____Numeral__Onatural_Mt__Code____Numeral__Onatural_J,type,
    iterat8892046348760725948atural: code_natural > ( code_natural > produc7822875418678951345atural > produc5835291356934675326atural ) > code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).

thf(sy_c_Random_Olog,type,
    log: code_natural > code_natural > code_natural ).

thf(sy_c_Random_Ominus__shift,type,
    minus_shift: code_natural > code_natural > code_natural > code_natural ).

thf(sy_c_Random_Onext,type,
    next: produc7822875418678951345atural > produc5835291356934675326atural ).

thf(sy_c_Random_Orange,type,
    range: code_natural > produc7822875418678951345atural > produc5835291356934675326atural ).

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_ORep__Rat,type,
    rep_Rat: rat > product_prod_int_int ).

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

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

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

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

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

thf(sy_c_Real_ORatreal,type,
    ratreal: rat > real ).

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,
    positive2: 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_Obounded__linear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V5970128139526366754l_real: ( real > real ) > $o ).

thf(sy_c_Real__Vector__Spaces_Olinear_001t__Real__Oreal_001t__Real__Oreal,type,
    real_V4572627801940501177l_real: ( real > real ) > $o ).

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

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

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

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

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

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

thf(sy_c_Relation_OId_001t__Nat__Onat,type,
    id_nat2: set_Pr1261947904930325089at_nat ).

thf(sy_c_Relation_Oantisym_001t__Nat__Onat,type,
    antisym_nat: set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Orefl__on_001t__Nat__Onat,type,
    refl_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Ototal__on_001t__Nat__Onat,type,
    total_on_nat: set_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Relation_Otrans_001t__Nat__Onat,type,
    trans_nat: set_Pr1261947904930325089at_nat > $o ).

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

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

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

thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Onatural,type,
    divide5121882707175180666atural: code_natural > code_natural > code_natural ).

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,
    divide_divide_real: real > real > real ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Onatural,type,
    dvd_dvd_Code_natural: code_natural > code_natural > $o ).

thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
    dvd_dvd_complex: complex > complex > $o ).

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,
    dvd_dvd_real: real > real > $o ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Ointeger,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(sy_c_Rings_Omodulo__class_Omodulo_001t__Code____Numeral__Onatural,type,
    modulo8411746178871703098atural: code_natural > code_natural > code_natural ).

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

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

thf(sy_c_Rings_Ounit__factor__class_Ounit__factor_001t__Nat__Onat,type,
    unit_f2748546683901255202or_nat: nat > nat ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Ointeger,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool_001t__Code____Numeral__Onatural,type,
    zero_n8403883297036319079atural: $o > code_natural ).

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

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

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

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

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

thf(sy_c_Series_Osuminf_001t__Real__Oreal,type,
    suminf_real: ( nat > real ) > real ).

thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
    summable_real: ( nat > real ) > $o ).

thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: ( nat > int ) > int > $o ).

thf(sy_c_Series_Osums_001t__Nat__Onat,type,
    sums_nat: ( nat > nat ) > nat > $o ).

thf(sy_c_Series_Osums_001t__Real__Oreal,type,
    sums_real: ( nat > real ) > real > $o ).

thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

thf(sy_c_Set_OCollect_001t__List__Olist_I_Eo_J,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Complex__Ocomplex_J,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    collec5989764272469232197st_nat: ( list_list_nat > $o ) > set_list_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
    collect_list_set_nat: ( list_set_nat > $o ) > set_list_set_nat ).

thf(sy_c_Set_OCollect_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Num__Onum,type,
    collect_num: ( num > $o ) > set_num ).

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

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

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

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

thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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

thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(sy_c_Set_Oimage_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    image_list_nat_nat: ( list_nat > nat ) > set_list_nat > set_nat ).

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    image_nat_list_nat: ( nat > list_nat ) > set_nat > set_list_nat ).

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_5846123807819985514at_nat: ( nat > product_prod_nat_nat ) > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Set__Oset_It__Nat__Onat_J,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

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

thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
    image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).

thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_2486076414777270412at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > set_nat ).

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

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

thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    image_1320371278474632150at_nat: ( sum_sum_nat_nat > nat ) > set_Sum_sum_nat_nat > set_nat ).

thf(sy_c_Set_Oinsert_001_Eo,type,
    insert_o: $o > set_o > set_o ).

thf(sy_c_Set_Oinsert_001t__Complex__Ocomplex,type,
    insert_complex: complex > set_complex > set_complex ).

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

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

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

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

thf(sy_c_Set_Oinsert_001t__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__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_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__Code____Numeral__Ointeger,type,
    set_fo1084959871951514735nteger: ( nat > code_integer > code_integer ) > nat > nat > code_integer > code_integer ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatMost_001t__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__Int__Oint,type,
    set_or6656581121297822940st_int: int > int > set_int ).

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__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_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

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

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

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

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

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

thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inl_nat_nat: nat > sum_sum_nat_nat ).

thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_Inr_nat_nat: nat > sum_sum_nat_nat ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Int__Oint_001t__Nat__Onat,type,
    sum_ca7763040182479039464nt_nat: ( nat > int ) > ( nat > int ) > sum_sum_nat_nat > int ).

thf(sy_c_Sum__Type_Osum_Ocase__sum_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
    sum_ca6763686470577984908at_nat: ( nat > nat ) > ( nat > nat ) > sum_sum_nat_nat > nat ).

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

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

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

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

thf(sy_c_Topological__Spaces_Otopological__space__class_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_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_Ocot_001t__Real__Oreal,type,
    cot_real: real > real ).

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

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

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

thf(sy_c_Transcendental_Olog,type,
    log2: real > real > real ).

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

thf(sy_c_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_Transitive__Closure_Otranclp_001t__Nat__Onat,type,
    transi2163837189807498211lp_nat: ( nat > nat > $o ) > nat > nat > $o ).

thf(sy_c_Typerep_Otyperep_OTyperep,type,
    typerep2: literal > list_typerep > typerep ).

thf(sy_c_Typerep_Otyperep_Osize__typerep,type,
    size_typerep: typerep > nat ).

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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__comp__shift_001t__Nat__Onat,type,
    vEBT_V2881884560877996034ft_nat: ( nat > nat > $o ) > option_nat > option_nat > $o ).

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__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_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__Code____Numeral__Ointeger,type,
    member_Code_integer: code_integer > set_Code_integer > $o ).

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

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

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

thf(sy_c_member_001t__List__Olist_It__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_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Complex__Ocomplex_J,type,
    member1046615901120239500omplex: produc648051720047351925omplex > set_Pr5421754520313593387omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
    member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_M_Eo_J,type,
    member6487239523555734774plex_o: produc6088675342482847199plex_o > set_Pr216032351708956309plex_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    member5793383173714906214omplex: produc4411394909380815293omplex > set_Pr5085853215250843933omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Int__Oint_J,type,
    member595073364599660772ex_int: produc6845221339535797307ex_int > set_Pr2254670189886740123ex_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Nat__Onat_J,type,
    member4772924384108857480ex_nat: produc1799700322190218207ex_nat > set_Pr4744753334818466879ex_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Real__Oreal_J,type,
    member47443559803733732x_real: produc8892588492097263291x_real > set_Pr1133549439701694107x_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__VEBT____Definitions__OVEBT_J,type,
    member1978952105866562066T_VEBT: produc7151440242714718331T_VEBT > set_Pr4085867452638698417T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Complex__Ocomplex_J,type,
    member8811922270175639012omplex: produc5838698208256999739omplex > set_Pr1846070511934368667omplex > $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__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member8253032945758599840st_int: produc1713839591385758857st_int > set_Pr5001190662893202239st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member1519744053835550788st_nat: produc4203922736317485613st_nat > set_Pr591367044826345187st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member1087064965665443052T_VEBT: produc1922972420619397443T_VEBT > set_Pr5170412164475753123T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Code____Numeral__Ointeger_J_Mt__List__Olist_It__Code____Numeral__Ointeger_J_J,type,
    member749217712838834276nteger: produc862207588354017979nteger > set_Pr7565137564259432987nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_I_Eo_J_J,type,
    member6688923169008879818list_o: produc149729814636038835list_o > set_Pr1150278048023938153list_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member5968030670617646438T_VEBT: produc872621073311890639T_VEBT > set_Pr1262583345697558789T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member6693912407220327184at_nat: produc6392793444374437607at_nat > set_Pr1542805901266377927at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    member4080735728053443344at_nat: produc424102278133772007at_nat > set_Pr4333006031979791559at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_I_Eo_J_J,type,
    member3126162362653435956list_o: produc3962069817607390347list_o > set_Pr7508168486584781291list_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Int__Oint_J_J,type,
    member3703241499402361532st_int: produc7831203938951381541st_int > set_Pr4080907618048478043st_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member6193324644334088288st_nat: produc1097915047028332489st_nat > set_Pr8894456036836396799st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__VEBT____Definitions__OVEBT_J_Mt__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $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__VEBT____Definitions__OVEBT_J,type,
    member8549952807677709168T_VEBT: produc8025551001238799321T_VEBT > set_Pr6167073792073659919T_VEBT > $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__Real__Oreal_M_Eo_J,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Complex__Ocomplex_J,type,
    member7358116576843751780omplex: produc6979889472282505531omplex > set_Pr6591433984475009307omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).

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

thf(sy_c_member_001t__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__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Complex__Ocomplex_J,type,
    member3207599676835851048omplex: produc8380087813684007313omplex > set_Pr216944050393469383omplex > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
    member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
    member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $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__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

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

thf(sy_v_a____,type,
    a: $o ).

thf(sy_v_b____,type,
    b: $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 (9514)
thf(fact_0_case4_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% case4(9)
thf(fact_1_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_2_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_3_option_Oinject,axiom,
    ! [X2: num,Y2: num] :
      ( ( ( some_num @ X2 )
        = ( some_num @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: code_integer,X2: code_integer,Y1: code_integer,Y2: code_integer] :
      ( ( ( produc1086072967326762835nteger @ X1 @ X2 )
        = ( produc1086072967326762835nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_prod_Oinject,axiom,
    ! [X1: product_prod_nat_nat,X2: product_prod_nat_nat,Y1: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ X1 @ X2 )
        = ( produc6161850002892822231at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_6_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_7_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_8_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_9_old_Oprod_Oinject,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ( produc1086072967326762835nteger @ A @ B )
        = ( produc1086072967326762835nteger @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_11_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_12_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_13_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_14_prod__decode__aux_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [K: nat,M: nat] :
          ( X
         != ( product_Pair_nat_nat @ K @ M ) ) ).

% prod_decode_aux.cases
thf(fact_15_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8923325533196201883nteger] :
      ~ ! [A3: code_integer,B3: code_integer] :
          ( Y
         != ( produc1086072967326762835nteger @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_16_old_Oprod_Oexhaust,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_17_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_18_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_19_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_20_surj__pair,axiom,
    ! [P: produc8923325533196201883nteger] :
    ? [X3: code_integer,Y3: code_integer] :
      ( P
      = ( produc1086072967326762835nteger @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_21_surj__pair,axiom,
    ! [P: produc859450856879609959at_nat] :
    ? [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( P
      = ( produc6161850002892822231at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_22_surj__pair,axiom,
    ! [P: produc3843707927480180839at_nat] :
    ? [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( P
      = ( produc2922128104949294807at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_23_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_24_surj__pair,axiom,
    ! [P: product_prod_int_int] :
    ? [X3: int,Y3: int] :
      ( P
      = ( product_Pair_int_int @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_25_prod__cases,axiom,
    ! [P2: produc8923325533196201883nteger > $o,P: produc8923325533196201883nteger] :
      ( ! [A3: code_integer,B3: code_integer] : ( P2 @ ( produc1086072967326762835nteger @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_26_prod__cases,axiom,
    ! [P2: produc859450856879609959at_nat > $o,P: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_27_prod__cases,axiom,
    ! [P2: produc3843707927480180839at_nat > $o,P: produc3843707927480180839at_nat] :
      ( ! [A3: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] : ( P2 @ ( produc2922128104949294807at_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_28_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_29_prod__cases,axiom,
    ! [P2: product_prod_int_int > $o,P: product_prod_int_int] :
      ( ! [A3: int,B3: int] : ( P2 @ ( product_Pair_int_int @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_30_Pair__inject,axiom,
    ! [A: code_integer,B: code_integer,A2: code_integer,B2: code_integer] :
      ( ( ( produc1086072967326762835nteger @ A @ B )
        = ( produc1086072967326762835nteger @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_31_Pair__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: product_prod_nat_nat,B2: product_prod_nat_nat] :
      ( ( ( produc6161850002892822231at_nat @ A @ B )
        = ( produc6161850002892822231at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_32_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_33_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_34_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_35_prod__cases3,axiom,
    ! [Y: produc859450856879609959at_nat] :
      ~ ! [A3: product_prod_nat_nat,B3: nat,C: nat] :
          ( Y
         != ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_36_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs )
             => ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).

% max_in_set_def
thf(fact_37_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs: set_nat,X4: nat] :
          ( ( member_nat @ X4 @ Xs )
          & ! [Y4: nat] :
              ( ( member_nat @ Y4 @ Xs )
             => ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% min_in_set_def
thf(fact_38_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% lesseq_shift
thf(fact_39_prod__induct3,axiom,
    ! [P2: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
      ( ! [A3: product_prod_nat_nat,B3: nat,C: nat] : ( P2 @ ( produc6161850002892822231at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_40_order__refl,axiom,
    ! [X: set_nat] : ( ord_less_eq_set_nat @ X @ X ) ).

% order_refl
thf(fact_41_order__refl,axiom,
    ! [X: rat] : ( ord_less_eq_rat @ X @ X ) ).

% order_refl
thf(fact_42_order__refl,axiom,
    ! [X: num] : ( ord_less_eq_num @ X @ X ) ).

% order_refl
thf(fact_43_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_44_order__refl,axiom,
    ! [X: int] : ( ord_less_eq_int @ X @ X ) ).

% order_refl
thf(fact_45_dual__order_Orefl,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% dual_order.refl
thf(fact_46_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_47_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_48_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_49_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_50_relChain__def,axiom,
    ( bNF_Ca8354645632395198811er_rat
    = ( ^ [R: set_Pr4811707699266497531nteger,As: code_integer > rat] :
        ! [I: code_integer,J: code_integer] :
          ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ I @ J ) @ R )
         => ( ord_less_eq_rat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_51_relChain__def,axiom,
    ( bNF_Ca333620267926924494at_rat
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > rat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_rat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_52_relChain__def,axiom,
    ( bNF_Ca1332973979827979050nt_rat
    = ( ^ [R: set_Pr958786334691620121nt_int,As: int > rat] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R )
         => ( ord_less_eq_rat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_53_relChain__def,axiom,
    ( bNF_Ca5547107478637473181er_num
    = ( ^ [R: set_Pr4811707699266497531nteger,As: code_integer > num] :
        ! [I: code_integer,J: code_integer] :
          ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ I @ J ) @ R )
         => ( ord_less_eq_num @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_54_relChain__def,axiom,
    ( bNF_Ca6749454151023974672at_num
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > num] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_num @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_55_relChain__def,axiom,
    ( bNF_Ca7748807862925029228nt_num
    = ( ^ [R: set_Pr958786334691620121nt_int,As: int > num] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R )
         => ( ord_less_eq_num @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_56_relChain__def,axiom,
    ( bNF_Ca8989775692481694547er_nat
    = ( ^ [R: set_Pr4811707699266497531nteger,As: code_integer > nat] :
        ! [I: code_integer,J: code_integer] :
          ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_57_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_58_relChain__def,axiom,
    ( bNF_Ca1968104039914474786nt_nat
    = ( ^ [R: set_Pr958786334691620121nt_int,As: int > nat] :
        ! [I: int,J: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ I @ J ) @ R )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_59_relChain__def,axiom,
    ( bNF_Ca8987285221972644271er_int
    = ( ^ [R: set_Pr4811707699266497531nteger,As: code_integer > int] :
        ! [I: code_integer,J: code_integer] :
          ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ I @ J ) @ R )
         => ( ord_less_eq_int @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_60_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X: produc4471711990508489141at_nat] :
      ~ ! [F: nat > nat > nat,A3: nat,B3: nat,Acc: nat] :
          ( X
         != ( produc3209952032786966637at_nat @ F @ ( produc487386426758144856at_nat @ A3 @ ( product_Pair_nat_nat @ B3 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_61_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_62_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_63_lift__Suc__mono__le,axiom,
    ! [F2: nat > set_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_set_nat @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_64_lift__Suc__mono__le,axiom,
    ! [F2: nat > rat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_rat @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_65_lift__Suc__mono__le,axiom,
    ! [F2: nat > num,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_num @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_66_lift__Suc__mono__le,axiom,
    ! [F2: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_67_lift__Suc__mono__le,axiom,
    ! [F2: nat > int,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_int @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_68_lift__Suc__antimono__le,axiom,
    ! [F2: nat > set_nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_set_nat @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_69_lift__Suc__antimono__le,axiom,
    ! [F2: nat > rat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_rat @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_70_lift__Suc__antimono__le,axiom,
    ! [F2: nat > num,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_num @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_71_lift__Suc__antimono__le,axiom,
    ! [F2: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_72_lift__Suc__antimono__le,axiom,
    ! [F2: nat > int,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F2 @ ( suc @ N3 ) ) @ ( F2 @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_int @ ( F2 @ N2 ) @ ( F2 @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_73_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_74_le__trans,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K2 )
       => ( ord_less_eq_nat @ I2 @ K2 ) ) ) ).

% le_trans
thf(fact_75__C1_C_I2_J,axiom,
    ( deg
    = ( suc @ zero_zero_nat ) ) ).

% "1"(2)
thf(fact_76_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_77_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_78_prod__encode__eq,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_encode @ X )
        = ( nat_prod_encode @ Y ) )
      = ( X = Y ) ) ).

% prod_encode_eq
thf(fact_79_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_80_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_81_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_82_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R2: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X3: nat] : ( R2 @ X3 @ X3 )
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( ( R2 @ X3 @ Y3 )
             => ( ( R2 @ Y3 @ Z )
               => ( R2 @ X3 @ Z ) ) )
         => ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
           => ( R2 @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_83_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P2 @ M2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M2 @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_84_mem__Collect__eq,axiom,
    ! [A: int,P2: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
    ! [A: complex,P2: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_86_mem__Collect__eq,axiom,
    ! [A: real,P2: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
    ! [A: list_nat,P2: list_nat > $o] :
      ( ( member_list_nat @ A @ ( collect_list_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_88_mem__Collect__eq,axiom,
    ! [A: set_nat,P2: set_nat > $o] :
      ( ( member_set_nat @ A @ ( collect_set_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_89_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A4: set_int] :
      ( ( collect_int
        @ ^ [X4: int] : ( member_int @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_91_Collect__mem__eq,axiom,
    ! [A4: set_complex] :
      ( ( collect_complex
        @ ^ [X4: complex] : ( member_complex @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_92_Collect__mem__eq,axiom,
    ! [A4: set_real] :
      ( ( collect_real
        @ ^ [X4: real] : ( member_real @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_93_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_94_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_95_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_96_Collect__cong,axiom,
    ! [P2: complex > $o,Q: complex > $o] :
      ( ! [X3: complex] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_complex @ P2 )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_97_Collect__cong,axiom,
    ! [P2: real > $o,Q: real > $o] :
      ( ! [X3: real] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_real @ P2 )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_98_Collect__cong,axiom,
    ! [P2: list_nat > $o,Q: list_nat > $o] :
      ( ! [X3: list_nat] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_list_nat @ P2 )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_99_Collect__cong,axiom,
    ! [P2: set_nat > $o,Q: set_nat > $o] :
      ( ! [X3: set_nat] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_set_nat @ P2 )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_100_Collect__cong,axiom,
    ! [P2: nat > $o,Q: nat > $o] :
      ( ! [X3: nat] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_nat @ P2 )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_101_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_102_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_103_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_104_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_105_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_106_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_107_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_108_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_109_order__antisym__conv,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_110_order__antisym__conv,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_111_order__antisym__conv,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_112_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_113_order__antisym__conv,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_114_linorder__le__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_115_linorder__le__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_116_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_117_linorder__le__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_118_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_119_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_120_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_121_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_122_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_123_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_124_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_125_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_126_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_127_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F2: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_128_ord__eq__le__subst,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_129_ord__eq__le__subst,axiom,
    ! [A: num,F2: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_130_ord__eq__le__subst,axiom,
    ! [A: nat,F2: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_131_ord__eq__le__subst,axiom,
    ! [A: int,F2: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_132_ord__eq__le__subst,axiom,
    ! [A: rat,F2: num > rat,B: num,C2: num] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_133_ord__eq__le__subst,axiom,
    ! [A: num,F2: num > num,B: num,C2: num] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_134_ord__eq__le__subst,axiom,
    ! [A: nat,F2: num > nat,B: num,C2: num] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_135_ord__eq__le__subst,axiom,
    ! [A: int,F2: num > int,B: num,C2: num] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_136_ord__eq__le__subst,axiom,
    ! [A: rat,F2: nat > rat,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_137_ord__eq__le__subst,axiom,
    ! [A: num,F2: nat > num,B: nat,C2: nat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_138_linorder__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_linear
thf(fact_139_linorder__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_linear
thf(fact_140_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_141_linorder__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_linear
thf(fact_142_order__eq__refl,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( X = Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_143_order__eq__refl,axiom,
    ! [X: rat,Y: rat] :
      ( ( X = Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_144_order__eq__refl,axiom,
    ! [X: num,Y: num] :
      ( ( X = Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_eq_refl
thf(fact_145_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_146_order__eq__refl,axiom,
    ! [X: int,Y: int] :
      ( ( X = Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_eq_refl
thf(fact_147_order__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_148_order__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_149_order__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_150_order__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_151_order__subst2,axiom,
    ! [A: num,B: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_152_order__subst2,axiom,
    ! [A: num,B: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_153_order__subst2,axiom,
    ! [A: num,B: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_154_order__subst2,axiom,
    ! [A: num,B: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_155_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_156_order__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > num,C2: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_157_order__subst1,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_158_order__subst1,axiom,
    ! [A: rat,F2: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_159_order__subst1,axiom,
    ! [A: rat,F2: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_160_order__subst1,axiom,
    ! [A: rat,F2: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_161_order__subst1,axiom,
    ! [A: num,F2: rat > num,B: rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_162_order__subst1,axiom,
    ! [A: num,F2: num > num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_163_order__subst1,axiom,
    ! [A: num,F2: nat > num,B: nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_164_order__subst1,axiom,
    ! [A: num,F2: int > num,B: int,C2: int] :
      ( ( ord_less_eq_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_165_order__subst1,axiom,
    ! [A: nat,F2: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_166_order__subst1,axiom,
    ! [A: nat,F2: num > nat,B: num,C2: num] :
      ( ( ord_less_eq_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_167_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z2: set_nat] : Y5 = Z2 )
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_168_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z2: rat] : Y5 = Z2 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_169_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z2: num] : Y5 = Z2 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_170_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_171_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_172_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_173_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_174_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_175_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_176_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_177_dual__order_Otrans,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ B )
       => ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_178_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_179_dual__order_Otrans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_eq_num @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_180_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_181_dual__order_Otrans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_182_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_183_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_184_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_185_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_186_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_187_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z2: set_nat] : Y5 = Z2 )
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A5 )
          & ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_188_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: rat,Z2: rat] : Y5 = Z2 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_189_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: num,Z2: num] : Y5 = Z2 )
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_190_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_191_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_192_linorder__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat,B3: rat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_193_linorder__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num,B3: num] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_194_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_195_linorder__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int,B3: int] :
            ( ( P2 @ B3 @ A3 )
           => ( P2 @ A3 @ B3 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_196_order__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z3 )
       => ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_197_order__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z3 )
       => ( ord_less_eq_rat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_198_order__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z3 )
       => ( ord_less_eq_num @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_199_order__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_200_order__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_eq_int @ X @ Z3 ) ) ) ).

% order_trans
thf(fact_201_order_Otrans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_202_order_Otrans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_203_order_Otrans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% order.trans
thf(fact_204_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_205_order_Otrans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% order.trans
thf(fact_206_order__antisym,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_207_order__antisym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_208_order__antisym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_209_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_210_order__antisym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_211_ord__le__eq__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_212_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_213_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_214_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_215_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_216_ord__eq__le__trans,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( A = B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_218_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_eq_num @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_219_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_220_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_221_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: set_nat,Z2: set_nat] : Y5 = Z2 )
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y4 )
          & ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_222_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: rat,Z2: rat] : Y5 = Z2 )
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_223_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: num,Z2: num] : Y5 = Z2 )
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_224_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_225_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_226_le__cases3,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ( ord_less_eq_rat @ X @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_rat @ Y @ X )
         => ~ ( ord_less_eq_rat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_rat @ X @ Z3 )
           => ~ ( ord_less_eq_rat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z3 @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z3 )
               => ~ ( ord_less_eq_rat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_rat @ Z3 @ X )
                 => ~ ( ord_less_eq_rat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_227_le__cases3,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ( ord_less_eq_num @ X @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_num @ Y @ X )
         => ~ ( ord_less_eq_num @ X @ Z3 ) )
       => ( ( ( ord_less_eq_num @ X @ Z3 )
           => ~ ( ord_less_eq_num @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_num @ Z3 @ Y )
             => ~ ( ord_less_eq_num @ Y @ X ) )
           => ( ( ( ord_less_eq_num @ Y @ Z3 )
               => ~ ( ord_less_eq_num @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_num @ Z3 @ X )
                 => ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_228_le__cases3,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_229_le__cases3,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ( ord_less_eq_int @ X @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_int @ Y @ X )
         => ~ ( ord_less_eq_int @ X @ Z3 ) )
       => ( ( ( ord_less_eq_int @ X @ Z3 )
           => ~ ( ord_less_eq_int @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_int @ Z3 @ Y )
             => ~ ( ord_less_eq_int @ Y @ X ) )
           => ( ( ( ord_less_eq_int @ Y @ Z3 )
               => ~ ( ord_less_eq_int @ Z3 @ X ) )
             => ~ ( ( ord_less_eq_int @ Z3 @ X )
                 => ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_230_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_231_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_232_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_233_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_234_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M5: nat] :
      ( ( P2 @ X )
     => ( ! [X3: nat] :
            ( ( P2 @ X3 )
           => ( ord_less_eq_nat @ X3 @ M5 ) )
       => ~ ! [M: nat] :
              ( ( P2 @ M )
             => ~ ! [X5: nat] :
                    ( ( P2 @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_235_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X3: nat] :
            ( ( P2 @ X3 )
            & ! [Y6: nat] :
                ( ( P2 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_236_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_237_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_238_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_239_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_240_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_241_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_242_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_243_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_244_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_245_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% not0_implies_Suc
thf(fact_246_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_247_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_248_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_249_zero__induct,axiom,
    ! [P2: nat > $o,K2: nat] :
      ( ( P2 @ K2 )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_250_diff__induct,axiom,
    ! [P2: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X3: nat] : ( P2 @ X3 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X3: nat,Y3: nat] :
              ( ( P2 @ X3 @ Y3 )
             => ( P2 @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_251_list__decode_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_252_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_253_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_254_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_255_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_256_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_257_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( P2 @ ( suc @ N3 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_258_case4_I12_J,axiom,
    vEBT_invar_vebt @ sa @ deg ).

% case4(12)
thf(fact_259_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_260_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_261_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_262_option_Osize_I4_J,axiom,
    ! [X2: num] :
      ( ( size_size_option_num @ ( some_num @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_263_vebt__buildup_Ocases,axiom,
    ! [X: nat] :
      ( ( X != zero_zero_nat )
     => ( ( X
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va: nat] :
              ( X
             != ( suc @ ( suc @ Va ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_264_exists__least__lemma,axiom,
    ! [P2: nat > $o] :
      ( ~ ( P2 @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P2 @ X_1 )
       => ? [N3: nat] :
            ( ~ ( P2 @ N3 )
            & ( P2 @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_265_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_266_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_267_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_268_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_269_zero__le,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).

% zero_le
thf(fact_270_divides__aux__eq,axiom,
    ! [Q2: code_integer,R3: code_integer] :
      ( ( unique5706413561485394159nteger @ ( produc1086072967326762835nteger @ Q2 @ R3 ) )
      = ( R3 = zero_z3403309356797280102nteger ) ) ).

% divides_aux_eq
thf(fact_271_divides__aux__eq,axiom,
    ! [Q2: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_272_divides__aux__eq,axiom,
    ! [Q2: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_273_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_274__C1_C_I1_J,axiom,
    ( sa
    = ( vEBT_Leaf @ a @ b ) ) ).

% "1"(1)
thf(fact_275_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_276_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_277_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_278_prod__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_prod_decode @ X )
        = ( nat_prod_decode @ Y ) )
      = ( X = Y ) ) ).

% prod_decode_eq
thf(fact_279_insert_H__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_VEBT_insert @ T @ X ) @ N ) ) ).

% insert'_pres_valid
thf(fact_280_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_281_mult__0__right,axiom,
    ! [M2: nat] :
      ( ( times_times_nat @ M2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_282_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_283_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_284_prod__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
      = N ) ).

% prod_decode_inverse
thf(fact_285_prod__encode__inverse,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
      = X ) ).

% prod_encode_inverse
thf(fact_286_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_287_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_288_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C2 ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% mult.left_commute
thf(fact_289_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C2 ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% mult.left_commute
thf(fact_290_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C2 ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% mult.left_commute
thf(fact_291_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C2 ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% mult.left_commute
thf(fact_292_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A5: real,B4: real] : ( times_times_real @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_293_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A5: rat,B4: rat] : ( times_times_rat @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_294_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( times_times_nat @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_295_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A5: int,B4: int] : ( times_times_int @ B4 @ A5 ) ) ) ).

% mult.commute
thf(fact_296_mult_Oassoc,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
      = ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_297_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_298_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_299_mult_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
      = ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% mult.assoc
thf(fact_300_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C2 )
      = ( times_times_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_301_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C2 )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_302_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_303_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C2 )
      = ( times_times_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_304_size__neq__size__imp__neq,axiom,
    ! [X: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_305_size__neq__size__imp__neq,axiom,
    ! [X: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X )
       != ( size_size_list_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_306_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_307_size__neq__size__imp__neq,axiom,
    ! [X: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X )
       != ( size_size_list_int @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_308_size__neq__size__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ( size_size_num @ X )
       != ( size_size_num @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_309_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_310_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_311_mult__le__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).

% mult_le_mono2
thf(fact_312_mult__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).

% mult_le_mono1
thf(fact_313_mult__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).

% mult_le_mono
thf(fact_314_le__square,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).

% le_square
thf(fact_315_le__cube,axiom,
    ! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).

% le_cube
thf(fact_316_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_317_zero__reorient,axiom,
    ! [X: complex] :
      ( ( zero_zero_complex = X )
      = ( X = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_318_zero__reorient,axiom,
    ! [X: real] :
      ( ( zero_zero_real = X )
      = ( X = zero_zero_real ) ) ).

% zero_reorient
thf(fact_319_zero__reorient,axiom,
    ! [X: rat] :
      ( ( zero_zero_rat = X )
      = ( X = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_320_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_321_zero__reorient,axiom,
    ! [X: int] :
      ( ( zero_zero_int = X )
      = ( X = zero_zero_int ) ) ).

% zero_reorient
thf(fact_322_mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( ( times_times_complex @ A @ C2 )
        = ( times_times_complex @ B @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_323_mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ( times_times_real @ A @ C2 )
        = ( times_times_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_324_mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C2 )
        = ( times_times_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_325_mult__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C2 )
        = ( times_times_nat @ B @ C2 ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_326_mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ( times_times_int @ A @ C2 )
        = ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_327_mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( ( times_times_complex @ C2 @ A )
        = ( times_times_complex @ C2 @ B ) )
      = ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_328_mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( times_times_real @ C2 @ A )
        = ( times_times_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_329_mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C2 @ A )
        = ( times_times_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_330_mult__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C2 @ A )
        = ( times_times_nat @ C2 @ B ) )
      = ( ( C2 = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_331_mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ( times_times_int @ C2 @ A )
        = ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_332_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_333_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_334_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_335_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_336_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_337_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_338_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_339_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_340_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_341_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_342_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_343_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_344_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_345_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_346_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_347_mul__shift,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = Z3 )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z3 ) ) ) ).

% mul_shift
thf(fact_348_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_349_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_350_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_351_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_352_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_353_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_354_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_355_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_356_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_357_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_358_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_359_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_360_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_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X22 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_369_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_370_VEBT_Osize_I4_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size(4)
thf(fact_371_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_372_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_373_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_374_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_375_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_376_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_377_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_378_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_379_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_380_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_381_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( times_times_complex @ A @ B )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_382_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_383_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_384_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_385_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_386_mult__left__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ C2 @ A )
          = ( times_times_complex @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_387_mult__left__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ C2 @ A )
          = ( times_times_real @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_388_mult__left__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ C2 @ A )
          = ( times_times_rat @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_389_mult__left__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ C2 @ A )
          = ( times_times_nat @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_390_mult__left__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ C2 @ A )
          = ( times_times_int @ C2 @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_391_mult__right__cancel,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C2 )
          = ( times_times_complex @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_392_mult__right__cancel,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ A @ C2 )
          = ( times_times_real @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_393_mult__right__cancel,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C2 )
          = ( times_times_rat @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_394_mult__right__cancel,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C2 )
          = ( times_times_nat @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_395_mult__right__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( ( times_times_int @ A @ C2 )
          = ( times_times_int @ B @ C2 ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_396_mult__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_397_mult__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_398_mult__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_399_mult__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_400_mult__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_401_mult__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_402_mult__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_403_mult__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_404_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_405_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_406_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_407_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_408_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_409_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_410_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_411_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_412_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_413_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_414_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_415_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_416_mult__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_417_mult__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_418_mult__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_419_mult__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_left_mono
thf(fact_420_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_421_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_422_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono_neg
thf(fact_423_mult__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_424_mult__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_425_mult__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_426_mult__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_right_mono
thf(fact_427_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_428_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_429_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_430_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_431_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_432_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_433_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_434_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_435_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_436_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_437_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_438_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_439_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_440_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_441_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A5: $o,B4: $o] :
            ( T
            = ( vEBT_Leaf @ A5 @ B4 ) ) ) ) ).

% deg1Leaf
thf(fact_442_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_443_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_444_nat__mult__eq__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( times_times_nat @ K2 @ M2 )
        = ( times_times_nat @ K2 @ N ) )
      = ( ( K2 = zero_zero_nat )
        | ( M2 = N ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_445_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_446_VEBT_Osize__gen_I2_J,axiom,
    ! [X21: $o,X22: $o] :
      ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
      = zero_zero_nat ) ).

% VEBT.size_gen(2)
thf(fact_447_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
     => ? [Info: option4927543243414619207at_nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info @ ( suc @ ( suc @ N ) ) @ TreeList @ S ) ) ) ).

% deg_SUcn_Node
thf(fact_448_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_449_deg__deg__n,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
     => ( Deg = N ) ) ).

% deg_deg_n
thf(fact_450_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_451_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_452_mult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% mult_1
thf(fact_453_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_454_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_455_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_456_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_457_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_458_mult_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.right_neutral
thf(fact_459_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_460_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_461_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_462_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_463_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_464_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_465_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_466_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_467_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_468_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_469_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_470_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_471_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_472_case4_I3_J,axiom,
    vEBT_invar_vebt @ summary2 @ m ).

% case4(3)
thf(fact_473_mult__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( C2
        = ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_left1
thf(fact_474_mult__cancel__left1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2
        = ( times_times_complex @ C2 @ B ) )
      = ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_475_mult__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( C2
        = ( times_times_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_476_mult__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2
        = ( times_times_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_477_mult__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_478_mult__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ( times_3573771949741848930nteger @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_left2
thf(fact_479_mult__cancel__left2,axiom,
    ! [C2: complex,A: complex] :
      ( ( ( times_times_complex @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_left2
thf(fact_480_mult__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ( times_times_real @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_left2
thf(fact_481_mult__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ( times_times_rat @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_482_mult__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ( times_times_int @ C2 @ A )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_left2
thf(fact_483_mult__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( C2
        = ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( B = one_one_Code_integer ) ) ) ).

% mult_cancel_right1
thf(fact_484_mult__cancel__right1,axiom,
    ! [C2: complex,B: complex] :
      ( ( C2
        = ( times_times_complex @ B @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_485_mult__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( C2
        = ( times_times_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_486_mult__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( C2
        = ( times_times_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_487_mult__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( C2
        = ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_488_mult__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_z3403309356797280102nteger )
        | ( A = one_one_Code_integer ) ) ) ).

% mult_cancel_right2
thf(fact_489_mult__cancel__right2,axiom,
    ! [A: complex,C2: complex] :
      ( ( ( times_times_complex @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_complex )
        | ( A = one_one_complex ) ) ) ).

% mult_cancel_right2
thf(fact_490_mult__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ( times_times_real @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_real )
        | ( A = one_one_real ) ) ) ).

% mult_cancel_right2
thf(fact_491_mult__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ( times_times_rat @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_492_mult__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ( times_times_int @ A @ C2 )
        = C2 )
      = ( ( C2 = zero_zero_int )
        | ( A = one_one_int ) ) ) ).

% mult_cancel_right2
thf(fact_493_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_494_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_495_nat__mult__less__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K2 )
        & ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_496_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_497_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_498_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_499_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_500_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_501_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_502_linorder__neqE__linordered__idom,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_503_linorder__neqE__linordered__idom,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_504_linorder__neqE__linordered__idom,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_505_lt__ex,axiom,
    ! [X: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X ) ).

% lt_ex
thf(fact_506_lt__ex,axiom,
    ! [X: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X ) ).

% lt_ex
thf(fact_507_lt__ex,axiom,
    ! [X: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).

% lt_ex
thf(fact_508_gt__ex,axiom,
    ! [X: real] :
    ? [X_12: real] : ( ord_less_real @ X @ X_12 ) ).

% gt_ex
thf(fact_509_gt__ex,axiom,
    ! [X: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X @ X_12 ) ).

% gt_ex
thf(fact_510_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_511_gt__ex,axiom,
    ! [X: int] :
    ? [X_12: int] : ( ord_less_int @ X @ X_12 ) ).

% gt_ex
thf(fact_512_dense,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ? [Z: real] :
          ( ( ord_less_real @ X @ Z )
          & ( ord_less_real @ Z @ Y ) ) ) ).

% dense
thf(fact_513_dense,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ? [Z: rat] :
          ( ( ord_less_rat @ X @ Z )
          & ( ord_less_rat @ Z @ Y ) ) ) ).

% dense
thf(fact_514_less__imp__neq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_515_less__imp__neq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_516_less__imp__neq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_517_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_518_less__imp__neq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_519_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_520_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_521_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_522_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_523_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_524_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_525_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_526_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_527_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_528_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_529_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_530_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_531_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_532_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_533_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_534_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X3: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X3 )
             => ( P2 @ Y6 ) )
         => ( P2 @ X3 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_535_antisym__conv3,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_real @ Y @ X )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_536_antisym__conv3,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_rat @ Y @ X )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_537_antisym__conv3,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_num @ Y @ X )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_538_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_539_antisym__conv3,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_int @ Y @ X )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_540_linorder__cases,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_541_linorder__cases,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_542_linorder__cases,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_543_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_544_linorder__cases,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_545_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_546_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_547_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_548_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_549_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_550_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_551_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_552_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_553_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_554_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_555_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N4: nat] :
          ( ( P4 @ N4 )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N4 )
             => ~ ( P4 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_556_linorder__less__wlog,axiom,
    ! [P2: real > real > $o,A: real,B: real] :
      ( ! [A3: real,B3: real] :
          ( ( ord_less_real @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: real] : ( P2 @ A3 @ A3 )
       => ( ! [A3: real,B3: real] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_557_linorder__less__wlog,axiom,
    ! [P2: rat > rat > $o,A: rat,B: rat] :
      ( ! [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: rat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: rat,B3: rat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_558_linorder__less__wlog,axiom,
    ! [P2: num > num > $o,A: num,B: num] :
      ( ! [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: num] : ( P2 @ A3 @ A3 )
       => ( ! [A3: num,B3: num] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_559_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_560_linorder__less__wlog,axiom,
    ! [P2: int > int > $o,A: int,B: int] :
      ( ! [A3: int,B3: int] :
          ( ( ord_less_int @ A3 @ B3 )
         => ( P2 @ A3 @ B3 ) )
     => ( ! [A3: int] : ( P2 @ A3 @ A3 )
       => ( ! [A3: int,B3: int] :
              ( ( P2 @ B3 @ A3 )
             => ( P2 @ A3 @ B3 ) )
         => ( P2 @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_561_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_562_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_563_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_564_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_565_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_566_not__less__iff__gr__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ( ord_less_real @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_567_not__less__iff__gr__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ( ord_less_rat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_568_not__less__iff__gr__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ( ord_less_num @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_569_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_570_not__less__iff__gr__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ( ord_less_int @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_571_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_572_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_573_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_574_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_575_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_576_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_577_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_578_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_579_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_580_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_581_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_582_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_583_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_584_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_585_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_586_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_587_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_588_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_589_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_590_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_591_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_592_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P2 @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P2 @ M3 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_593_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_594_linorder__neqE,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
     => ( ~ ( ord_less_real @ X @ Y )
       => ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_595_linorder__neqE,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
     => ( ~ ( ord_less_rat @ X @ Y )
       => ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_596_linorder__neqE,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
     => ( ~ ( ord_less_num @ X @ Y )
       => ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_597_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_598_linorder__neqE,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
     => ( ~ ( ord_less_int @ X @ Y )
       => ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_599_order__less__asym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_asym
thf(fact_600_order__less__asym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_asym
thf(fact_601_order__less__asym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_asym
thf(fact_602_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_603_order__less__asym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_asym
thf(fact_604_linorder__neq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( X != Y )
      = ( ( ord_less_real @ X @ Y )
        | ( ord_less_real @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_605_linorder__neq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( X != Y )
      = ( ( ord_less_rat @ X @ Y )
        | ( ord_less_rat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_606_linorder__neq__iff,axiom,
    ! [X: num,Y: num] :
      ( ( X != Y )
      = ( ( ord_less_num @ X @ Y )
        | ( ord_less_num @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_607_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_608_linorder__neq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( X != Y )
      = ( ( ord_less_int @ X @ Y )
        | ( ord_less_int @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_609_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_610_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_611_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_612_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_613_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_614_order__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_615_order__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_616_order__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_617_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_618_order__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_trans
thf(fact_619_ord__eq__less__subst,axiom,
    ! [A: real,F2: real > real,B: real,C2: real] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_620_ord__eq__less__subst,axiom,
    ! [A: rat,F2: real > rat,B: real,C2: real] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_621_ord__eq__less__subst,axiom,
    ! [A: num,F2: real > num,B: real,C2: real] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_622_ord__eq__less__subst,axiom,
    ! [A: nat,F2: real > nat,B: real,C2: real] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_623_ord__eq__less__subst,axiom,
    ! [A: int,F2: real > int,B: real,C2: real] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_624_ord__eq__less__subst,axiom,
    ! [A: real,F2: rat > real,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_625_ord__eq__less__subst,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_626_ord__eq__less__subst,axiom,
    ! [A: num,F2: rat > num,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_627_ord__eq__less__subst,axiom,
    ! [A: nat,F2: rat > nat,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_628_ord__eq__less__subst,axiom,
    ! [A: int,F2: rat > int,B: rat,C2: rat] :
      ( ( A
        = ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_629_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_630_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_631_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: real > num,C2: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_632_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: real > nat,C2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_633_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F2: real > int,C2: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_634_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_635_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_636_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_637_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_638_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F2: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F2 @ B )
          = C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_639_order__less__irrefl,axiom,
    ! [X: real] :
      ~ ( ord_less_real @ X @ X ) ).

% order_less_irrefl
thf(fact_640_order__less__irrefl,axiom,
    ! [X: rat] :
      ~ ( ord_less_rat @ X @ X ) ).

% order_less_irrefl
thf(fact_641_order__less__irrefl,axiom,
    ! [X: num] :
      ~ ( ord_less_num @ X @ X ) ).

% order_less_irrefl
thf(fact_642_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_643_order__less__irrefl,axiom,
    ! [X: int] :
      ~ ( ord_less_int @ X @ X ) ).

% order_less_irrefl
thf(fact_644_order__less__subst1,axiom,
    ! [A: real,F2: real > real,B: real,C2: real] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_645_order__less__subst1,axiom,
    ! [A: real,F2: rat > real,B: rat,C2: rat] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_646_order__less__subst1,axiom,
    ! [A: real,F2: num > real,B: num,C2: num] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_647_order__less__subst1,axiom,
    ! [A: real,F2: nat > real,B: nat,C2: nat] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_648_order__less__subst1,axiom,
    ! [A: real,F2: int > real,B: int,C2: int] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_649_order__less__subst1,axiom,
    ! [A: rat,F2: real > rat,B: real,C2: real] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_650_order__less__subst1,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_651_order__less__subst1,axiom,
    ! [A: rat,F2: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_652_order__less__subst1,axiom,
    ! [A: rat,F2: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_653_order__less__subst1,axiom,
    ! [A: rat,F2: int > rat,B: int,C2: int] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_654_order__less__subst2,axiom,
    ! [A: real,B: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_655_order__less__subst2,axiom,
    ! [A: real,B: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_656_order__less__subst2,axiom,
    ! [A: real,B: real,F2: real > num,C2: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_657_order__less__subst2,axiom,
    ! [A: real,B: real,F2: real > nat,C2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_658_order__less__subst2,axiom,
    ! [A: real,B: real,F2: real > int,C2: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_659_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_660_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_661_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > num,C2: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_662_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_663_order__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > int,C2: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_664_order__less__not__sym,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_665_order__less__not__sym,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_666_order__less__not__sym,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_667_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_668_order__less__not__sym,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_669_order__less__imp__triv,axiom,
    ! [X: real,Y: real,P2: $o] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_real @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_670_order__less__imp__triv,axiom,
    ! [X: rat,Y: rat,P2: $o] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_671_order__less__imp__triv,axiom,
    ! [X: num,Y: num,P2: $o] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_num @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_672_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_673_order__less__imp__triv,axiom,
    ! [X: int,Y: int,P2: $o] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_int @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_674_linorder__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
      | ( X = Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_675_linorder__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
      | ( X = Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_676_linorder__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
      | ( X = Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_677_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_678_linorder__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
      | ( X = Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_679_order__less__imp__not__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_680_order__less__imp__not__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_681_order__less__imp__not__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_682_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_683_order__less__imp__not__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_684_order__less__imp__not__eq2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_685_order__less__imp__not__eq2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_686_order__less__imp__not__eq2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_687_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_688_order__less__imp__not__eq2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_689_order__less__imp__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ~ ( ord_less_real @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_690_order__less__imp__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ~ ( ord_less_rat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_691_order__less__imp__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ~ ( ord_less_num @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_692_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_693_order__less__imp__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ~ ( ord_less_int @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_694_less__numeral__extra_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% less_numeral_extra(4)
thf(fact_695_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_696_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_697_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_698_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_699_one__reorient,axiom,
    ! [X: code_integer] :
      ( ( one_one_Code_integer = X )
      = ( X = one_one_Code_integer ) ) ).

% one_reorient
thf(fact_700_one__reorient,axiom,
    ! [X: complex] :
      ( ( one_one_complex = X )
      = ( X = one_one_complex ) ) ).

% one_reorient
thf(fact_701_one__reorient,axiom,
    ! [X: real] :
      ( ( one_one_real = X )
      = ( X = one_one_real ) ) ).

% one_reorient
thf(fact_702_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_703_one__reorient,axiom,
    ! [X: int] :
      ( ( one_one_int = X )
      = ( X = one_one_int ) ) ).

% one_reorient
thf(fact_704_less__numeral__extra_I1_J,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% less_numeral_extra(1)
thf(fact_705_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_706_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_707_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_708_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_709_not__one__less__zero,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_less_zero
thf(fact_710_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_711_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_712_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_713_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_714_zero__less__one,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one
thf(fact_715_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_716_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_717_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_718_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_719_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > real,N: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F2 @ N ) @ ( F2 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_720_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > rat,N: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_rat @ ( F2 @ N ) @ ( F2 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_721_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > num,N: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_num @ ( F2 @ N ) @ ( F2 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_722_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > nat,N: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_723_lift__Suc__mono__less__iff,axiom,
    ! [F2: nat > int,N: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F2 @ N ) @ ( F2 @ M2 ) )
        = ( ord_less_nat @ N @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_724_lift__Suc__mono__less,axiom,
    ! [F2: nat > real,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_real @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_725_lift__Suc__mono__less,axiom,
    ! [F2: nat > rat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_rat @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_726_lift__Suc__mono__less,axiom,
    ! [F2: nat > num,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_num @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_727_lift__Suc__mono__less,axiom,
    ! [F2: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_nat @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_728_lift__Suc__mono__less,axiom,
    ! [F2: nat > int,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_int @ ( F2 @ N ) @ ( F2 @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_729_less__1__mult,axiom,
    ! [M2: code_integer,N: code_integer] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ M2 )
     => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_730_less__1__mult,axiom,
    ! [M2: real,N: real] :
      ( ( ord_less_real @ one_one_real @ M2 )
     => ( ( ord_less_real @ one_one_real @ N )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_731_less__1__mult,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_rat @ one_one_rat @ M2 )
     => ( ( ord_less_rat @ one_one_rat @ N )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_732_less__1__mult,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ one_one_nat @ M2 )
     => ( ( ord_less_nat @ one_one_nat @ N )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_733_less__1__mult,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ one_one_int @ M2 )
     => ( ( ord_less_int @ one_one_int @ N )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M2 @ N ) ) ) ) ).

% less_1_mult
thf(fact_734_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,D: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
      = ( D = one_one_nat ) ) ).

% VEBT_internal.valid'.simps(1)
thf(fact_735_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_736_order__le__imp__less__or__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_737_order__le__imp__less__or__eq,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_738_order__le__imp__less__or__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_739_order__le__imp__less__or__eq,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_740_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_741_order__le__imp__less__or__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_742_linorder__le__less__linear,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
      | ( ord_less_real @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_743_linorder__le__less__linear,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
      | ( ord_less_rat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_744_linorder__le__less__linear,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
      | ( ord_less_num @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_745_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_746_linorder__le__less__linear,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
      | ( ord_less_int @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_747_order__less__le__subst2,axiom,
    ! [A: real,B: real,F2: real > real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_748_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > real,C2: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_749_order__less__le__subst2,axiom,
    ! [A: num,B: num,F2: num > real,C2: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_750_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > real,C2: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_751_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: int > real,C2: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_752_order__less__le__subst2,axiom,
    ! [A: real,B: real,F2: real > rat,C2: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_753_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_754_order__less__le__subst2,axiom,
    ! [A: num,B: num,F2: num > rat,C2: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_755_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F2: nat > rat,C2: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_756_order__less__le__subst2,axiom,
    ! [A: int,B: int,F2: int > rat,C2: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_757_order__less__le__subst1,axiom,
    ! [A: real,F2: rat > real,B: rat,C2: rat] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_758_order__less__le__subst1,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_759_order__less__le__subst1,axiom,
    ! [A: num,F2: rat > num,B: rat,C2: rat] :
      ( ( ord_less_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_760_order__less__le__subst1,axiom,
    ! [A: nat,F2: rat > nat,B: rat,C2: rat] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_761_order__less__le__subst1,axiom,
    ! [A: int,F2: rat > int,B: rat,C2: rat] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_762_order__less__le__subst1,axiom,
    ! [A: real,F2: num > real,B: num,C2: num] :
      ( ( ord_less_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_763_order__less__le__subst1,axiom,
    ! [A: rat,F2: num > rat,B: num,C2: num] :
      ( ( ord_less_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_764_order__less__le__subst1,axiom,
    ! [A: num,F2: num > num,B: num,C2: num] :
      ( ( ord_less_num @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_765_order__less__le__subst1,axiom,
    ! [A: nat,F2: num > nat,B: num,C2: num] :
      ( ( ord_less_nat @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_766_order__less__le__subst1,axiom,
    ! [A: int,F2: num > int,B: num,C2: num] :
      ( ( ord_less_int @ A @ ( F2 @ B ) )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_767_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > real,C2: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_768_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_769_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > num,C2: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_770_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_771_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F2: rat > int,C2: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_772_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: num > real,C2: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_773_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: num > rat,C2: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_774_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: num > num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_num @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_num @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_775_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: num > nat,C2: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_nat @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_776_order__le__less__subst2,axiom,
    ! [A: num,B: num,F2: num > int,C2: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F2 @ B ) @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_eq_num @ X3 @ Y3 )
             => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_777_order__le__less__subst1,axiom,
    ! [A: real,F2: real > real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_778_order__le__less__subst1,axiom,
    ! [A: real,F2: rat > real,B: rat,C2: rat] :
      ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_779_order__le__less__subst1,axiom,
    ! [A: real,F2: num > real,B: num,C2: num] :
      ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_780_order__le__less__subst1,axiom,
    ! [A: real,F2: nat > real,B: nat,C2: nat] :
      ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_781_order__le__less__subst1,axiom,
    ! [A: real,F2: int > real,B: int,C2: int] :
      ( ( ord_less_eq_real @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_782_order__le__less__subst1,axiom,
    ! [A: rat,F2: real > rat,B: real,C2: real] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_real @ B @ C2 )
       => ( ! [X3: real,Y3: real] :
              ( ( ord_less_real @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_783_order__le__less__subst1,axiom,
    ! [A: rat,F2: rat > rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ! [X3: rat,Y3: rat] :
              ( ( ord_less_rat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_784_order__le__less__subst1,axiom,
    ! [A: rat,F2: num > rat,B: num,C2: num] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_num @ B @ C2 )
       => ( ! [X3: num,Y3: num] :
              ( ( ord_less_num @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_785_order__le__less__subst1,axiom,
    ! [A: rat,F2: nat > rat,B: nat,C2: nat] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X3: nat,Y3: nat] :
              ( ( ord_less_nat @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_786_order__le__less__subst1,axiom,
    ! [A: rat,F2: int > rat,B: int,C2: int] :
      ( ( ord_less_eq_rat @ A @ ( F2 @ B ) )
     => ( ( ord_less_int @ B @ C2 )
       => ( ! [X3: int,Y3: int] :
              ( ( ord_less_int @ X3 @ Y3 )
             => ( ord_less_rat @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F2 @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_787_order__less__le__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_788_order__less__le__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ Y @ Z3 )
       => ( ord_less_set_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_789_order__less__le__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_790_order__less__le__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_791_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_792_order__less__le__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_793_order__le__less__trans,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ord_less_real @ Y @ Z3 )
       => ( ord_less_real @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_794_order__le__less__trans,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_set_nat @ Y @ Z3 )
       => ( ord_less_set_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_795_order__le__less__trans,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_rat @ Y @ Z3 )
       => ( ord_less_rat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_796_order__le__less__trans,axiom,
    ! [X: num,Y: num,Z3: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_less_num @ Y @ Z3 )
       => ( ord_less_num @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_797_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_798_order__le__less__trans,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_int @ Y @ Z3 )
       => ( ord_less_int @ X @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_799_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_800_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_801_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_802_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_803_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_804_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_805_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_806_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_807_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_808_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_809_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_810_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_811_order__less__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_812_order__less__imp__le,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_813_order__less__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_814_order__less__imp__le,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_815_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_816_order__less__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_817_linorder__not__less,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_real @ X @ Y ) )
      = ( ord_less_eq_real @ Y @ X ) ) ).

% linorder_not_less
thf(fact_818_linorder__not__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X @ Y ) )
      = ( ord_less_eq_rat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_819_linorder__not__less,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_num @ X @ Y ) )
      = ( ord_less_eq_num @ Y @ X ) ) ).

% linorder_not_less
thf(fact_820_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_821_linorder__not__less,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_int @ X @ Y ) )
      = ( ord_less_eq_int @ Y @ X ) ) ).

% linorder_not_less
thf(fact_822_linorder__not__le,axiom,
    ! [X: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X @ Y ) )
      = ( ord_less_real @ Y @ X ) ) ).

% linorder_not_le
thf(fact_823_linorder__not__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X @ Y ) )
      = ( ord_less_rat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_824_linorder__not__le,axiom,
    ! [X: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X @ Y ) )
      = ( ord_less_num @ Y @ X ) ) ).

% linorder_not_le
thf(fact_825_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_826_linorder__not__le,axiom,
    ! [X: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X @ Y ) )
      = ( ord_less_int @ Y @ X ) ) ).

% linorder_not_le
thf(fact_827_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_eq_real @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_828_order__less__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_829_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_830_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_831_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_832_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ( X4 != Y4 ) ) ) ) ).

% order_less_le
thf(fact_833_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_real @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_834_order__le__less,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( ord_less_set_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_835_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_rat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_836_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_num @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_837_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_nat @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_838_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_int @ X4 @ Y4 )
          | ( X4 = Y4 ) ) ) ) ).

% order_le_less
thf(fact_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_eq_real @ B4 @ A5 )
          & ~ ( ord_less_eq_real @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_852_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_set_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_853_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ~ ( ord_less_eq_rat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_854_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ~ ( ord_less_eq_num @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_855_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_856_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ~ ( ord_less_eq_int @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_857_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_858_dual__order_Ostrict__trans2,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ B )
       => ( ord_less_set_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_859_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_860_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_861_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_862_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_863_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C2 @ B )
       => ( ord_less_real @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_864_dual__order_Ostrict__trans1,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_set_nat @ C2 @ B )
       => ( ord_less_set_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_865_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ B )
       => ( ord_less_rat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_866_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C2 @ B )
       => ( ord_less_num @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_867_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_868_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C2 @ B )
       => ( ord_less_int @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_869_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_eq_real @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_870_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( ord_less_eq_set_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_871_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_eq_rat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_872_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_eq_num @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_873_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_874_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_eq_int @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_875_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B4: real,A5: real] :
          ( ( ord_less_real @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_876_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( ord_less_set_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_877_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_less_rat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_878_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_less_num @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_879_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_880_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_less_int @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_881_dense__le__bounded,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( ord_less_real @ X @ Y )
     => ( ! [W: real] :
            ( ( ord_less_real @ X @ W )
           => ( ( ord_less_real @ W @ Y )
             => ( ord_less_eq_real @ W @ Z3 ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_le_bounded
thf(fact_882_dense__le__bounded,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_rat @ X @ Y )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X @ W )
           => ( ( ord_less_rat @ W @ Y )
             => ( ord_less_eq_rat @ W @ Z3 ) ) )
       => ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% dense_le_bounded
thf(fact_883_dense__ge__bounded,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ Z3 @ X )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z3 @ W )
           => ( ( ord_less_real @ W @ X )
             => ( ord_less_eq_real @ Y @ W ) ) )
       => ( ord_less_eq_real @ Y @ Z3 ) ) ) ).

% dense_ge_bounded
thf(fact_884_dense__ge__bounded,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ Z3 @ X )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z3 @ W )
           => ( ( ord_less_rat @ W @ X )
             => ( ord_less_eq_rat @ Y @ W ) ) )
       => ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% dense_ge_bounded
thf(fact_885_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
          & ~ ( ord_less_eq_real @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_886_order_Ostrict__iff__not,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_set_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_887_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ~ ( ord_less_eq_rat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_888_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ~ ( ord_less_eq_num @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_889_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_890_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ~ ( ord_less_eq_int @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_891_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_892_order_Ostrict__trans2,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_893_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_894_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_895_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_896_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_897_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_898_order_Ostrict__trans1,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_set_nat @ B @ C2 )
       => ( ord_less_set_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_899_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_900_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C2: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C2 )
       => ( ord_less_num @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_901_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_902_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_903_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_eq_real @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_904_order_Ostrict__iff__order,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_eq_set_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_905_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_906_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_eq_num @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_907_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_908_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_eq_int @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_909_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] :
          ( ( ord_less_real @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_910_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( ord_less_set_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_911_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_less_rat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_912_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_less_num @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_913_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_914_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_less_int @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_915_not__le__imp__less,axiom,
    ! [Y: real,X: real] :
      ( ~ ( ord_less_eq_real @ Y @ X )
     => ( ord_less_real @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_916_not__le__imp__less,axiom,
    ! [Y: rat,X: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X )
     => ( ord_less_rat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_917_not__le__imp__less,axiom,
    ! [Y: num,X: num] :
      ( ~ ( ord_less_eq_num @ Y @ X )
     => ( ord_less_num @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_918_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_919_not__le__imp__less,axiom,
    ! [Y: int,X: int] :
      ( ~ ( ord_less_eq_int @ Y @ X )
     => ( ord_less_int @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_920_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X4: real,Y4: real] :
          ( ( ord_less_eq_real @ X4 @ Y4 )
          & ~ ( ord_less_eq_real @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_921_less__le__not__le,axiom,
    ( ord_less_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( ord_less_eq_set_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_set_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_922_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( ord_less_eq_rat @ X4 @ Y4 )
          & ~ ( ord_less_eq_rat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_923_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X4: num,Y4: num] :
          ( ( ord_less_eq_num @ X4 @ Y4 )
          & ~ ( ord_less_eq_num @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_924_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_925_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X4: int,Y4: int] :
          ( ( ord_less_eq_int @ X4 @ Y4 )
          & ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_926_dense__le,axiom,
    ! [Y: real,Z3: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ X3 @ Y )
         => ( ord_less_eq_real @ X3 @ Z3 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_le
thf(fact_927_dense__le,axiom,
    ! [Y: rat,Z3: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ X3 @ Y )
         => ( ord_less_eq_rat @ X3 @ Z3 ) )
     => ( ord_less_eq_rat @ Y @ Z3 ) ) ).

% dense_le
thf(fact_928_dense__ge,axiom,
    ! [Z3: real,Y: real] :
      ( ! [X3: real] :
          ( ( ord_less_real @ Z3 @ X3 )
         => ( ord_less_eq_real @ Y @ X3 ) )
     => ( ord_less_eq_real @ Y @ Z3 ) ) ).

% dense_ge
thf(fact_929_dense__ge,axiom,
    ! [Z3: rat,Y: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_rat @ Z3 @ X3 )
         => ( ord_less_eq_rat @ Y @ X3 ) )
     => ( ord_less_eq_rat @ Y @ Z3 ) ) ).

% dense_ge
thf(fact_930_antisym__conv2,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ Y )
     => ( ( ~ ( ord_less_real @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_931_antisym__conv2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ~ ( ord_less_set_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_932_antisym__conv2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ~ ( ord_less_rat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_933_antisym__conv2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ~ ( ord_less_num @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_934_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_935_antisym__conv2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ~ ( ord_less_int @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_936_antisym__conv1,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_937_antisym__conv1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ~ ( ord_less_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_938_antisym__conv1,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_939_antisym__conv1,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ( ord_less_eq_num @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_940_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_941_antisym__conv1,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_942_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_943_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_944_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_945_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_946_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_947_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_948_leI,axiom,
    ! [X: real,Y: real] :
      ( ~ ( ord_less_real @ X @ Y )
     => ( ord_less_eq_real @ Y @ X ) ) ).

% leI
thf(fact_949_leI,axiom,
    ! [X: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X @ Y )
     => ( ord_less_eq_rat @ Y @ X ) ) ).

% leI
thf(fact_950_leI,axiom,
    ! [X: num,Y: num] :
      ( ~ ( ord_less_num @ X @ Y )
     => ( ord_less_eq_num @ Y @ X ) ) ).

% leI
thf(fact_951_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_952_leI,axiom,
    ! [X: int,Y: int] :
      ( ~ ( ord_less_int @ X @ Y )
     => ( ord_less_eq_int @ Y @ X ) ) ).

% leI
thf(fact_953_leD,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ Y @ X )
     => ~ ( ord_less_real @ X @ Y ) ) ).

% leD
thf(fact_954_leD,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ~ ( ord_less_set_nat @ X @ Y ) ) ).

% leD
thf(fact_955_leD,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ~ ( ord_less_rat @ X @ Y ) ) ).

% leD
thf(fact_956_leD,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ~ ( ord_less_num @ X @ Y ) ) ).

% leD
thf(fact_957_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_958_leD,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ~ ( ord_less_int @ X @ Y ) ) ).

% leD
thf(fact_959_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_960_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_961_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_962_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_963_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_964_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_965_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_966_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_967_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_968_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] :
            ( ( J2
              = ( suc @ I3 ) )
           => ( P2 @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( P2 @ ( suc @ I3 ) )
               => ( P2 @ I3 ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_969_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J3: nat,K: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( ( ord_less_nat @ J3 @ K )
               => ( ( P2 @ I3 @ J3 )
                 => ( ( P2 @ J3 @ K )
                   => ( P2 @ I3 @ K ) ) ) ) )
         => ( P2 @ I2 @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_970_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ J2 @ K2 )
       => ( ord_less_nat @ ( suc @ I2 ) @ K2 ) ) ) ).

% less_trans_Suc
thf(fact_971_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_972_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_973_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M7: nat] :
            ( ( M2
              = ( suc @ M7 ) )
            & ( ord_less_nat @ N @ M7 ) ) ) ) ).

% Suc_less_eq2
thf(fact_974_All__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ N )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P2 @ I ) ) ) ) ).

% All_less_Suc
thf(fact_975_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_976_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_977_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ N )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P2 @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_978_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_979_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_980_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_981_Suc__lessE,axiom,
    ! [I2: nat,K2: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K2 )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I2 @ J3 )
           => ( K2
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_982_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_983_Nat_OlessE,axiom,
    ! [I2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ K2 )
     => ( ( K2
         != ( suc @ I2 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( K2
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_984_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X22 ) ) ).

% VEBT.distinct(1)
thf(fact_985_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X222: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X222 ) ) ) ).

% VEBT.exhaust
thf(fact_986_le__numeral__extra_I4_J,axiom,
    ord_le3102999989581377725nteger @ one_one_Code_integer @ one_one_Code_integer ).

% le_numeral_extra(4)
thf(fact_987_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_988_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_989_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_990_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_991_zero__neq__one,axiom,
    zero_z3403309356797280102nteger != one_one_Code_integer ).

% zero_neq_one
thf(fact_992_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_993_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_994_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_995_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_996_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_997_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_998_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_999_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1000_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1001_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1002_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1003_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P2 @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P2 @ M3 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_1004_mult_Ocomm__neutral,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ one_one_Code_integer )
      = A ) ).

% mult.comm_neutral
thf(fact_1005_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_1006_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_1007_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_1008_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_1009_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_1010_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ one_one_Code_integer @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_1011_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_1012_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_1013_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_1014_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_1015_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_1016_less__mono__imp__le__mono,axiom,
    ! [F2: nat > nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F2 @ I3 ) @ ( F2 @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1017_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_1018_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_1019_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N4: nat] :
          ( ( ord_less_nat @ M6 @ N4 )
          | ( M6 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1020_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_1021_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M6 @ N4 )
          & ( M6 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_1022_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_1023_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_1024_nat__mult__eq__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ( times_times_nat @ K2 @ M2 )
          = ( times_times_nat @ K2 @ N ) )
        = ( M2 = N ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_1025_nat__mult__less__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_mult_less_cancel1
thf(fact_1026_mult__less__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1027_mult__less__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1028_mult__less__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1029_mult__less__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_1030_mult__less__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1031_mult__less__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1032_mult__less__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1033_mult__less__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_1034_mult__less__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ C2 )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1035_mult__less__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ one_one_real ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ one_one_real @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1036_mult__less__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1037_mult__less__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ one_one_int ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ one_one_int @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_1038_mult__less__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le3102999989581377725nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1039_mult__less__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_real @ C2 @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1040_mult__less__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1041_mult__less__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_1042_mult__le__cancel__right2,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1043_mult__le__cancel__right2,axiom,
    ! [A: real,C2: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1044_mult__le__cancel__right2,axiom,
    ! [A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1045_mult__le__cancel__right2,axiom,
    ! [A: int,C2: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_1046_mult__le__cancel__right1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ B @ C2 ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1047_mult__le__cancel__right1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1048_mult__le__cancel__right1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1049_mult__le__cancel__right1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_1050_mult__le__cancel__left2,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ C2 @ A ) @ C2 )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1051_mult__le__cancel__left2,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ one_one_real ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1052_mult__le__cancel__left2,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1053_mult__le__cancel__left2,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ C2 )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ one_one_int ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_1054_mult__le__cancel__left1,axiom,
    ! [C2: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ ( times_3573771949741848930nteger @ C2 @ B ) )
      = ( ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ C2 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ B ) )
        & ( ( ord_le6747313008572928689nteger @ C2 @ zero_z3403309356797280102nteger )
         => ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1055_mult__le__cancel__left1,axiom,
    ! [C2: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1056_mult__le__cancel__left1,axiom,
    ! [C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1057_mult__le__cancel__left1,axiom,
    ! [C2: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_1058_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1059_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1060_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_1061_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_1062_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_1063_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_1064_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1065_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1066_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_1067_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_1068_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_1069_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_1070_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_1071_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_1072_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_1073_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_1074_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_1075_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1076_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1077_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1078_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_1079_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_1080_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_1081_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_1082_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_1083_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1084_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1085_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_1086_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1087_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1088_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1089_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_1090_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1091_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1092_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1093_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_1094_mult__less__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1095_mult__less__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1096_mult__less__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_1097_mult__less__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1098_mult__less__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1099_mult__less__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_1100_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1101_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1102_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_1103_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1104_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1105_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1106_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_1107_mult__less__cancel__left__disj,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1108_mult__less__cancel__left__disj,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1109_mult__less__cancel__left__disj,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_1110_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1111_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1112_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C2 @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_1113_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1114_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1115_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1116_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% mult_strict_right_mono
thf(fact_1117_mult__less__cancel__right__disj,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C2 @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1118_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C2 @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1119_mult__less__cancel__right__disj,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C2 @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_1120_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1121_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1122_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1123_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1124_not__one__le__zero,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ).

% not_one_le_zero
thf(fact_1125_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1126_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1127_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1128_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1129_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_1130_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_1131_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_1132_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_1133_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_1134_zero__less__one__class_Ozero__le__one,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ one_one_Code_integer ).

% zero_less_one_class.zero_le_one
thf(fact_1135_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_1136_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_1137_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_1138_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_1139_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_1140_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J: nat] :
            ( ( M2
              = ( suc @ J ) )
            & ( ord_less_nat @ J @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1141_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M: nat] :
          ( N
          = ( suc @ M ) ) ) ).

% gr0_implies_Suc
thf(fact_1142_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P2 @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1143_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M6: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1144_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P2 @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1145_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_1146_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1147_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_1148_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_1149_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_1150_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ J2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J2 )
               => ( ( P2 @ ( suc @ N3 ) )
                 => ( P2 @ N3 ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% inc_induct
thf(fact_1151_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ I2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J2 )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( suc @ N3 ) ) ) ) )
         => ( P2 @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1152_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_1153_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1154_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_1155_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_1156_mult__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ I2 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).

% mult_less_mono1
thf(fact_1157_mult__less__mono2,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_nat @ ( times_times_nat @ K2 @ I2 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).

% mult_less_mono2
thf(fact_1158_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1159_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_1160_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1161_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1162_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1163_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_1164_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1165_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1166_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1167_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_1168_mult__right__le__imp__le,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_1169_mult__right__le__imp__le,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_1170_mult__right__le__imp__le,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_1171_mult__right__le__imp__le,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_1172_mult__left__le__imp__le,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_1173_mult__left__le__imp__le,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_1174_mult__left__le__imp__le,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C2 )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_1175_mult__left__le__imp__le,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C2 )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_1176_mult__le__cancel__left__pos,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1177_mult__le__cancel__left__pos,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1178_mult__le__cancel__left__pos,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_1179_mult__le__cancel__left__neg,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1180_mult__le__cancel__left__neg,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1181_mult__le__cancel__left__neg,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_1182_mult__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1183_mult__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1184_mult__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_1185_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1186_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1187_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1188_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_1189_mult__right__less__imp__less,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_1190_mult__right__less__imp__less,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_1191_mult__right__less__imp__less,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_1192_mult__right__less__imp__less,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_1193_mult__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1194_mult__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1195_mult__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C2 )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C2 @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_1196_mult__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
           => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1197_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
           => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1198_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
           => ( ord_less_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1199_mult__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
           => ( ord_less_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_1200_mult__left__less__imp__less,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_1201_mult__left__less__imp__less,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_1202_mult__left__less__imp__less,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_1203_mult__left__less__imp__less,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C2 )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_1204_mult__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1205_mult__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1206_mult__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_1207_mult__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1208_mult__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1209_mult__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C2 )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C2 @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_1210_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) @ Ux ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X3 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_1211_mult__left__le,axiom,
    ! [C2: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ C2 @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_1212_mult__left__le,axiom,
    ! [C2: real,A: real] :
      ( ( ord_less_eq_real @ C2 @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_1213_mult__left__le,axiom,
    ! [C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_1214_mult__left__le,axiom,
    ! [C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_1215_mult__left__le,axiom,
    ! [C2: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C2 ) @ A ) ) ) ).

% mult_left_le
thf(fact_1216_mult__le__one,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ) ).

% mult_le_one
thf(fact_1217_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_1218_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_1219_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_1220_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_1221_mult__right__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1222_mult__right__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1223_mult__right__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1224_mult__right__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).

% mult_right_le_one_le
thf(fact_1225_mult__left__le__one__le,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
       => ( ( ord_le3102999989581377725nteger @ Y @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( times_3573771949741848930nteger @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1226_mult__left__le__one__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1227_mult__left__le__one__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1228_mult__left__le__one__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).

% mult_left_le_one_le
thf(fact_1229_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1230_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_1231_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_1232_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_1233_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y4: nat,X4: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% greater_shift
thf(fact_1234_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y4: nat] : ( vEBT_VEBT_less @ ( some_nat @ X4 ) @ ( some_nat @ Y4 ) ) ) ) ).

% less_shift
thf(fact_1235_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_1236_field__le__mult__one__interval,axiom,
    ! [X: real,Y: real] :
      ( ! [Z: real] :
          ( ( ord_less_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ Y ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_1237_field__le__mult__one__interval,axiom,
    ! [X: rat,Y: rat] :
      ( ! [Z: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z )
         => ( ( ord_less_rat @ Z @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z @ X ) @ Y ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_1238_case4_I2_J,axiom,
    ! [S2: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ S2 @ m )
     => ( ( ( vEBT_VEBT_set_vebt @ summary2 )
          = ( vEBT_VEBT_set_vebt @ S2 ) )
       => ( S2 = summary2 ) ) ) ).

% case4(2)
thf(fact_1239_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% maxt_corr
thf(fact_1240_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X ) ) ) ) ).

% maxt_sound
thf(fact_1241_mint__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X ) ) ) ).

% mint_corr
thf(fact_1242_mint__sound,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X ) ) ) ) ).

% mint_sound
thf(fact_1243_mult__le__cancel__iff2,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ X ) @ ( times_times_real @ Z3 @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_1244_mult__le__cancel__iff2,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X ) @ ( times_times_rat @ Z3 @ Y ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_1245_mult__le__cancel__iff2,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z3 @ X ) @ ( times_times_int @ Z3 @ Y ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_1246_mult__le__cancel__iff1,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_1247_mult__le__cancel__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ Y @ Z3 ) )
        = ( ord_less_eq_rat @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_1248_mult__le__cancel__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_eq_int @ X @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_1249_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_1250_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_1251_case4_I5_J,axiom,
    m = na ).

% case4(5)
thf(fact_1252_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_1253_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_1254_mult__less__iff1,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( ( ord_less_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ Z3 ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1255_mult__less__iff1,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z3 )
     => ( ( ord_less_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ Y @ Z3 ) )
        = ( ord_less_rat @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1256_mult__less__iff1,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z3 )
     => ( ( ord_less_int @ ( times_times_int @ X @ Z3 ) @ ( times_times_int @ Y @ Z3 ) )
        = ( ord_less_int @ X @ Y ) ) ) ).

% mult_less_iff1
thf(fact_1257_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1258_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux2 @ Uy2 @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1259_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Maxi: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ X @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_1260_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N: nat,Mini: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X )
         => ( ord_less_eq_nat @ Mini @ X ) ) ) ) ).

% mint_corr_help
thf(fact_1261_buildup__nothing__in__leaf,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_leaf
thf(fact_1262_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_1263_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_1264_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_1265_VEBT__internal_Oinsert_H_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ X3 ) ) ) ).

% VEBT_internal.insert'.cases
thf(fact_1266_member__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ T @ X )
        = ( member_nat @ X @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_1267_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: 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 @ F2 @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F2 @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1268_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F2 @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F2 @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1269_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F2: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F2 @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F2 @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1270_buildup__gives__empty,axiom,
    ! [N: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_1271_pred__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ Y @ X )
        & ! [Z4: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z4 )
              & ( ord_less_nat @ Z4 @ X ) )
           => ( ord_less_eq_nat @ Z4 @ Y ) ) ) ) ).

% pred_member
thf(fact_1272_succ__member,axiom,
    ! [T: vEBT_VEBT,X: nat,Y: nat] :
      ( ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Y )
      = ( ( vEBT_vebt_member @ T @ Y )
        & ( ord_less_nat @ X @ Y )
        & ! [Z4: nat] :
            ( ( ( vEBT_vebt_member @ T @ Z4 )
              & ( ord_less_nat @ X @ Z4 ) )
           => ( ord_less_eq_nat @ Y @ Z4 ) ) ) ) ).

% succ_member
thf(fact_1273_case4_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% case4(6)
thf(fact_1274_case4_I1_J,axiom,
    ! [X5: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList2 ) )
     => ( ( vEBT_invar_vebt @ X5 @ na )
        & ! [Xa: vEBT_VEBT] :
            ( ( vEBT_invar_vebt @ Xa @ na )
           => ( ( ( vEBT_VEBT_set_vebt @ X5 )
                = ( vEBT_VEBT_set_vebt @ Xa ) )
             => ( Xa = X5 ) ) ) ) ) ).

% case4(1)
thf(fact_1275_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_1276_bot_Oextremum__uniqueI,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
     => ( A = bot_bot_set_real ) ) ).

% bot.extremum_uniqueI
thf(fact_1277_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_1278_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_1279_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_1280_bot_Oextremum__unique,axiom,
    ! [A: set_real] :
      ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
      = ( A = bot_bot_set_real ) ) ).

% bot.extremum_unique
thf(fact_1281_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_1282_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_1283_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_1284_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_1285_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_1286_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_1287_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_1288_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_1289_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_1290_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_1291_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_1292_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_1293_bot_Onot__eq__extremum,axiom,
    ! [A: set_real] :
      ( ( A != bot_bot_set_real )
      = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1294_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_1295_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
    ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw2 ) @ Ux2 ) ).

% VEBT_internal.naive_member.simps(2)
thf(fact_1296_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_1297_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N )
     => ( ( vEBT_vebt_member @ Tree @ X )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X )
          | ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).

% member_valid_both_member_options
thf(fact_1298_vebt__member_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X ) ).

% vebt_member.simps(4)
thf(fact_1299_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( ( ( X = zero_zero_nat )
         => A )
        & ( ( X != zero_zero_nat )
         => ( ( ( X = one_one_nat )
             => B )
            & ( X = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_1300_vebt__member_Osimps_I3_J,axiom,
    ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz ) @ X ) ).

% vebt_member.simps(3)
thf(fact_1301_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_1302_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_1303_buildup__nothing__in__min__max,axiom,
    ! [N: nat,X: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).

% buildup_nothing_in_min_max
thf(fact_1304_dele__member__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_vebt_member @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_vebt_member @ T @ Y ) ) ) ) ).

% dele_member_cont_corr
thf(fact_1305_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw ) )
     => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va2 @ Vb2 ) @ X3 ) )
         => ( ! [Mi2: nat,Ma2: nat,V: nat,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ X3 ) )
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ X3 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_1306_case4_I8_J,axiom,
    ( ( mi = ma )
   => ! [X5: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList2 ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ).

% case4(8)
thf(fact_1307_even__odd__cases,axiom,
    ! [X: nat] :
      ( ! [N3: nat] :
          ( X
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_1308_delete__pres__valid,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( vEBT_invar_vebt @ ( vEBT_vebt_delete @ T @ X ) @ N ) ) ).

% delete_pres_valid
thf(fact_1309_maxbmo,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( ( vEBT_vebt_maxt @ T )
        = ( some_nat @ X ) )
     => ( vEBT_V8194947554948674370ptions @ T @ X ) ) ).

% maxbmo
thf(fact_1310_dele__bmo__cont__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_delete @ T @ X ) @ Y )
        = ( ( X != Y )
          & ( vEBT_V8194947554948674370ptions @ T @ Y ) ) ) ) ).

% dele_bmo_cont_corr
thf(fact_1311_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
        = ( vEBT_vebt_member @ T @ X ) ) ) ).

% both_member_options_equiv_member
thf(fact_1312_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X )
       => ( vEBT_vebt_member @ T @ X ) ) ) ).

% valid_member_both_member_options
thf(fact_1313_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_1314_add__right__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1315_add__right__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1316_add__right__cancel,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1317_add__right__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add_right_cancel
thf(fact_1318_add__left__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1319_add__left__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1320_add__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1321_add__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add_left_cancel
thf(fact_1322_mi__eq__ma__no__ch,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi = Ma )
       => ( ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
             => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) )
          & ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1323_add__le__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1324_add__le__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1325_add__le__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1326_add__le__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_1327_add__le__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1328_add__le__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1329_add__le__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1330_add__le__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_1331_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_1332_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_1333_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_1334_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_1335_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_1336_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_1337_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_1338_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_1339_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_1340_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_1341_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_1342_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_1343_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_1344_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_1345_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_1346_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_1347_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_1348_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_1349_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ B @ A ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_1350_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_1351_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_1352_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_1353_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_1354_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_1355_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_1356_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_1357_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_1358_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_1359_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_1360_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y ) )
      = ( ( X = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_1361_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_1362_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_1363_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_1364_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_1365_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_1366_add__less__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1367_add__less__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1368_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1369_add__less__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_1370_add__less__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1371_add__less__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1372_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1373_add__less__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_1374_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_1375_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_1376_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_1377_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_1378_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_1379_not__Some__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( ! [Y4: product_prod_nat_nat] :
            ( X
           != ( some_P7363390416028606310at_nat @ Y4 ) ) )
      = ( X = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_1380_not__Some__eq,axiom,
    ! [X: option_nat] :
      ( ( ! [Y4: nat] :
            ( X
           != ( some_nat @ Y4 ) ) )
      = ( X = none_nat ) ) ).

% not_Some_eq
thf(fact_1381_not__Some__eq,axiom,
    ! [X: option_num] :
      ( ( ! [Y4: num] :
            ( X
           != ( some_num @ Y4 ) ) )
      = ( X = none_num ) ) ).

% not_Some_eq
thf(fact_1382_not__None__eq,axiom,
    ! [X: option4927543243414619207at_nat] :
      ( ( X != none_P5556105721700978146at_nat )
      = ( ? [Y4: product_prod_nat_nat] :
            ( X
            = ( some_P7363390416028606310at_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1383_not__None__eq,axiom,
    ! [X: option_nat] :
      ( ( X != none_nat )
      = ( ? [Y4: nat] :
            ( X
            = ( some_nat @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1384_not__None__eq,axiom,
    ! [X: option_num] :
      ( ( X != none_num )
      = ( ? [Y4: num] :
            ( X
            = ( some_num @ Y4 ) ) ) ) ).

% not_None_eq
thf(fact_1385_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_1386_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_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_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_1398_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_1399_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_1400_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_1401_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_1402_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_1403_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_1404_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_1405_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_1406_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_1407_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_1408_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_1409_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_1410_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_1411_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_1412_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_1413_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_1414_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_1415_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_1416_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_1417_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_1418_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_1419_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_1420_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_1421_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_1422_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_1423_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_1424_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_1425_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_1426_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_1427_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_1428_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_1429_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_1430_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_1431_add__right__imp__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1432_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1433_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1434_add__right__imp__eq,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
     => ( B = C2 ) ) ).

% add_right_imp_eq
thf(fact_1435_add__left__imp__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1436_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1437_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1438_add__left__imp__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
     => ( B = C2 ) ) ).

% add_left_imp_eq
thf(fact_1439_add_Oleft__commute,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C2 ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1440_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C2 ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1441_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1442_add_Oleft__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.left_commute
thf(fact_1443_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_1444_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_1445_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_1446_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_1447_add_Oright__cancel,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1448_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1449_add_Oright__cancel,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C2 @ A ) )
      = ( B = C2 ) ) ).

% add.right_cancel
thf(fact_1450_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1451_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1452_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C2 ) )
      = ( B = C2 ) ) ).

% add.left_cancel
thf(fact_1453_add_Oassoc,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1454_add_Oassoc,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1455_add_Oassoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1456_add_Oassoc,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add.assoc
thf(fact_1457_group__cancel_Oadd2,axiom,
    ! [B5: real,K2: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( plus_plus_real @ A @ B5 )
        = ( plus_plus_real @ K2 @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1458_group__cancel_Oadd2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( plus_plus_rat @ A @ B5 )
        = ( plus_plus_rat @ K2 @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1459_group__cancel_Oadd2,axiom,
    ! [B5: nat,K2: nat,B: nat,A: nat] :
      ( ( B5
        = ( plus_plus_nat @ K2 @ B ) )
     => ( ( plus_plus_nat @ A @ B5 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1460_group__cancel_Oadd2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( plus_plus_int @ A @ B5 )
        = ( plus_plus_int @ K2 @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_1461_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_1462_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_1463_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_1464_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_1465_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_real @ I2 @ K2 )
        = ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1466_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_rat @ I2 @ K2 )
        = ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1467_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I2 @ K2 )
        = ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1468_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( K2 = L ) )
     => ( ( plus_plus_int @ I2 @ K2 )
        = ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_1469_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_1470_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_1471_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% is_num_normalize(1)
thf(fact_1472_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1473_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1474_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1475_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_1476_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1477_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
    ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz ) ).

% VEBT_internal.membermima.simps(2)
thf(fact_1478_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1479_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1480_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1481_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_1482_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1483_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1484_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1485_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_1486_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J2 )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1487_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1488_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1489_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_1490_add__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1491_add__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1492_add__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1493_add__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_1494_add__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1495_add__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1496_add__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1497_add__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_left_mono
thf(fact_1498_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C: nat] :
            ( B
           != ( plus_plus_nat @ A @ C ) ) ) ).

% less_eqE
thf(fact_1499_add__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1500_add__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1501_add__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1502_add__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_right_mono
thf(fact_1503_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
        ? [C3: nat] :
          ( B4
          = ( plus_plus_nat @ A5 @ C3 ) ) ) ) ).

% le_iff_add
thf(fact_1504_add__le__imp__le__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1505_add__le__imp__le__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1506_add__le__imp__le__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1507_add__le__imp__le__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_1508_add__le__imp__le__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1509_add__le__imp__le__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1510_add__le__imp__le__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1511_add__le__imp__le__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_1512_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_1513_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_1514_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_1515_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_1516_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_1517_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_1518_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1519_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_1520_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1521_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1522_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1523_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1524_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1525_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1526_add__less__imp__less__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1527_add__less__imp__less__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1528_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1529_add__less__imp__less__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_1530_add__less__imp__less__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1531_add__less__imp__less__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1532_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1533_add__less__imp__less__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_1534_add__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1535_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1536_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1537_add__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_1538_add__strict__left__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1539_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1540_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1541_add__strict__left__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_1542_add__strict__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1543_add__strict__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1544_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1545_add__strict__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_1546_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1547_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1548_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1549_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( K2 = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_1550_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( I2 = J2 )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1551_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( I2 = J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1552_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( I2 = J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1553_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( I2 = J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_1554_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J2 )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1555_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1556_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1557_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_1558_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1559_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1560_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_1561_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1562_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1563_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_1564_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1565_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1566_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1567_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% comm_semiring_class.distrib
thf(fact_1568_distrib__left,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_1569_distrib__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_1570_distrib__left,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_1571_distrib__left,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% distrib_left
thf(fact_1572_distrib__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C2 )
      = ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_1573_distrib__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C2 )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_1574_distrib__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_1575_distrib__right,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% distrib_right
thf(fact_1576_combine__common__factor,axiom,
    ! [A: real,E: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1577_combine__common__factor,axiom,
    ! [A: rat,E: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1578_combine__common__factor,axiom,
    ! [A: nat,E: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1579_combine__common__factor,axiom,
    ! [A: int,E: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C2 ) ) ).

% combine_common_factor
thf(fact_1580_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1581_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: nat] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1582_combine__options__cases,axiom,
    ! [X: option4927543243414619207at_nat,P2: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_P5556105721700978146at_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: product_prod_nat_nat,B3: num] :
              ( ( X
                = ( some_P7363390416028606310at_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1583_combine__options__cases,axiom,
    ! [X: option_nat,P2: option_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: nat,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1584_combine__options__cases,axiom,
    ! [X: option_nat,P2: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1585_combine__options__cases,axiom,
    ! [X: option_nat,P2: option_nat > option_num > $o,Y: option_num] :
      ( ( ( X = none_nat )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: nat,B3: num] :
              ( ( X
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1586_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: product_prod_nat_nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1587_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option_nat > $o,Y: option_nat] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: nat] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1588_combine__options__cases,axiom,
    ! [X: option_num,P2: option_num > option_num > $o,Y: option_num] :
      ( ( ( X = none_num )
       => ( P2 @ X @ Y ) )
     => ( ( ( Y = none_num )
         => ( P2 @ X @ Y ) )
       => ( ! [A3: num,B3: num] :
              ( ( X
                = ( some_num @ A3 ) )
             => ( ( Y
                  = ( some_num @ B3 ) )
               => ( P2 @ X @ Y ) ) )
         => ( P2 @ X @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_1589_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1590_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X6: option_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1591_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_all
thf(fact_1592_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X4: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1593_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X6: option_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X4: nat] : ( P4 @ ( some_nat @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1594_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X4: num] : ( P4 @ ( some_num @ X4 ) ) ) ) ) ).

% split_option_ex
thf(fact_1595_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1596_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1597_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_1598_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X2: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X2 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_1599_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_1600_option_OdiscI,axiom,
    ! [Option: option_num,X2: num] :
      ( ( Option
        = ( some_num @ X2 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_1601_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_1602_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_1603_option_Odistinct_I1_J,axiom,
    ! [X2: num] :
      ( none_num
     != ( some_num @ X2 ) ) ).

% option.distinct(1)
thf(fact_1604_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
            ( X
           != ( produc2899441246263362727at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A3: product_prod_nat_nat,B3: product_prod_nat_nat] :
              ( X
             != ( produc2899441246263362727at_nat @ F @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A3 ) @ ( some_P7363390416028606310at_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1605_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw: nat > nat > nat,V: nat] :
            ( X
           != ( produc8929957630744042906on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V ) @ none_nat ) ) )
       => ~ ! [F: nat > nat > nat,A3: nat,B3: nat] :
              ( X
             != ( produc8929957630744042906on_nat @ F @ ( produc5098337634421038937on_nat @ ( some_nat @ A3 ) @ ( some_nat @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1606_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw: num > num > num,V: num] :
            ( X
           != ( produc5778274026573060048on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V ) @ none_num ) ) )
       => ~ ! [F: num > num > num,A3: num,B3: num] :
              ( X
             != ( produc5778274026573060048on_num @ F @ ( produc8585076106096196333on_num @ ( some_num @ A3 ) @ ( some_num @ B3 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1607_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw: product_prod_nat_nat > product_prod_nat_nat > $o,V: product_prod_nat_nat] :
            ( X
           != ( produc3994169339658061776at_nat @ Uw @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F: product_prod_nat_nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
              ( X
             != ( produc3994169339658061776at_nat @ F @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X3 ) @ ( some_P7363390416028606310at_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1608_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw: nat > nat > $o,V: nat] :
            ( X
           != ( produc4035269172776083154on_nat @ Uw @ ( produc5098337634421038937on_nat @ ( some_nat @ V ) @ none_nat ) ) )
       => ~ ! [F: nat > nat > $o,X3: nat,Y3: nat] :
              ( X
             != ( produc4035269172776083154on_nat @ F @ ( produc5098337634421038937on_nat @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1609_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw: num > num > $o,V: num] :
            ( X
           != ( produc3576312749637752826on_num @ Uw @ ( produc8585076106096196333on_num @ ( some_num @ V ) @ none_num ) ) )
       => ~ ! [F: num > num > $o,X3: num,Y3: num] :
              ( X
             != ( produc3576312749637752826on_num @ F @ ( produc8585076106096196333on_num @ ( some_num @ X3 ) @ ( some_num @ Y3 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1610_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_1611_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_1612_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_1613_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_1614_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1615_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_1616_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).

% trans_less_add2
thf(fact_1617_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).

% trans_less_add1
thf(fact_1618_add__less__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% add_less_mono1
thf(fact_1619_not__add__less2,axiom,
    ! [J2: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_1620_not__add__less1,axiom,
    ! [I2: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).

% not_add_less1
thf(fact_1621_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_1622_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 )
     => ( ord_less_nat @ I2 @ K2 ) ) ).

% add_lessD1
thf(fact_1623_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_1624_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_1625_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_1626_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_1627_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_1628_le__Suc__ex,axiom,
    ! [K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ K2 @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K2 @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_1629_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K2 @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_1630_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% add_le_mono1
thf(fact_1631_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1632_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).

% trans_le_add2
thf(fact_1633_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1634_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_1635_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_1636_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J2: nat,K2: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ K2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J2 ) @ U ) @ K2 ) ) ).

% left_add_mult_distrib
thf(fact_1637_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o,Uw2: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw2 ) ).

% VEBT_internal.membermima.simps(1)
thf(fact_1638_add__decreasing,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C2 @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1639_add__decreasing,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C2 @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1640_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1641_add__decreasing,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C2 @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_1642_add__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1643_add__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1644_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1645_add__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_1646_add__decreasing2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1647_add__decreasing2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1648_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1649_add__decreasing2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_1650_add__increasing2,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1651_add__increasing2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1652_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1653_add__increasing2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_1654_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_1655_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_1656_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_1657_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_1658_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_1659_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_1660_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_1661_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_1662_add__nonneg__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1663_add__nonneg__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1664_add__nonneg__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1665_add__nonneg__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1666_add__nonpos__eq__0__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X @ Y )
            = zero_zero_real )
          = ( ( X = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1667_add__nonpos__eq__0__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X @ Y )
            = zero_zero_rat )
          = ( ( X = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1668_add__nonpos__eq__0__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X @ Y )
            = zero_zero_nat )
          = ( ( X = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1669_add__nonpos__eq__0__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X @ Y )
            = zero_zero_int )
          = ( ( X = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1670_add__less__le__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1671_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1672_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1673_add__less__le__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_1674_add__le__less__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C2 @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1675_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1676_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1677_add__le__less__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C2 @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_1678_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J2 )
        & ( ord_less_eq_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1679_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J2 )
        & ( ord_less_eq_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1680_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J2 )
        & ( ord_less_eq_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1681_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J2 )
        & ( ord_less_eq_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_1682_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J2: real,K2: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J2 )
        & ( ord_less_real @ K2 @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K2 ) @ ( plus_plus_real @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1683_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J2: rat,K2: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J2 )
        & ( ord_less_rat @ K2 @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K2 ) @ ( plus_plus_rat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1684_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J2: nat,K2: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1685_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J2: int,K2: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J2 )
        & ( ord_less_int @ K2 @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K2 ) @ ( plus_plus_int @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_1686_add__less__zeroD,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1687_add__less__zeroD,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1688_add__less__zeroD,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1689_pos__add__strict,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1690_pos__add__strict,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1691_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1692_pos__add__strict,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_1693_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1694_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_1695_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_1696_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_1697_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_1698_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_1699_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_1700_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_1701_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_1702_add__mono1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ B @ one_one_Code_integer ) ) ) ).

% add_mono1
thf(fact_1703_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_1704_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_1705_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_1706_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_1707_less__add__one,axiom,
    ! [A: code_integer] : ( ord_le6747313008572928689nteger @ A @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) ) ).

% less_add_one
thf(fact_1708_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_1709_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_1710_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_1711_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_1712_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_1713_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_1714_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1715_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_1716_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_1717_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N4: nat] :
        ? [K3: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1718_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_1719_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I2 @ K )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_1720_mono__nat__linear__lb,axiom,
    ! [F2: nat > nat,M2: nat,K2: nat] :
      ( ! [M: nat,N3: nat] :
          ( ( ord_less_nat @ M @ N3 )
         => ( ord_less_nat @ ( F2 @ M ) @ ( F2 @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F2 @ M2 ) @ K2 ) @ ( F2 @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1721_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_1722_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1723_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1724_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1725_VEBT__internal_OminNull_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ( X
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_1726_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_1727_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_1728_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_1729_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y != none_P5556105721700978146at_nat ) )
       => ( ( ? [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 @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1730_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: num > num > num,Xa2: option_num,Xb: option_num,Y: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_num )
         => ( Y != none_num ) )
       => ( ( ? [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 @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1731_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X @ Xa2 @ Xb )
        = Y )
     => ( ( ( Xa2 = none_nat )
         => ( Y != none_nat ) )
       => ( ( ? [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 @ ( X @ A3 @ B3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1732_field__le__epsilon,axiom,
    ! [X: real,Y: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ X @ ( plus_plus_real @ Y @ E2 ) ) )
     => ( ord_less_eq_real @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_1733_field__le__epsilon,axiom,
    ! [X: rat,Y: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ X @ ( plus_plus_rat @ Y @ E2 ) ) )
     => ( ord_less_eq_rat @ X @ Y ) ) ).

% field_le_epsilon
thf(fact_1734_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_1735_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_1736_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_1737_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_1738_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_1739_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_1740_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_1741_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_1742_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_1743_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_1744_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_1745_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_1746_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_1747_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_1748_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_1749_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_1750_add__strict__increasing,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1751_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1752_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1753_add__strict__increasing,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_1754_add__strict__increasing2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C2 )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1755_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C2 )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1756_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1757_add__strict__increasing2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C2 )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_1758_sum__squares__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1759_sum__squares__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1760_sum__squares__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_1761_not__sum__squares__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_1762_not__sum__squares__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_1763_not__sum__squares__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_1764_discrete,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [A5: code_integer] : ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ A5 @ one_one_Code_integer ) ) ) ) ).

% discrete
thf(fact_1765_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A5 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_1766_discrete,axiom,
    ( ord_less_int
    = ( ^ [A5: int] : ( ord_less_eq_int @ ( plus_plus_int @ A5 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_1767_zero__less__two,axiom,
    ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer ) ).

% zero_less_two
thf(fact_1768_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_1769_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_1770_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1771_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_1772_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1773_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1774_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1775_convex__bound__le,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ X @ A )
     => ( ( ord_le3102999989581377725nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le3102999989581377725nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1776_convex__bound__le,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_eq_real @ X @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1777_convex__bound__le,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1778_convex__bound__le,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_1779_vebt__mint_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( A3
                 => ( Y
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A3
                 => ( ( B3
                     => ( Y
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_1780_vebt__maxt_Oelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( B3
                 => ( Y
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B3
                 => ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
           => ( Y != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
               => ( Y
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_1781_vebt__mint_Ocases,axiom,
    ! [X: vEBT_VEBT] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( vEBT_Leaf @ A3 @ B3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
            ( X
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_1782_convex__bound__lt,axiom,
    ! [X: code_integer,A: code_integer,Y: code_integer,U: code_integer,V2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X @ A )
     => ( ( ord_le6747313008572928689nteger @ Y @ A )
       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ U )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ V2 )
           => ( ( ( plus_p5714425477246183910nteger @ U @ V2 )
                = one_one_Code_integer )
             => ( ord_le6747313008572928689nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ U @ X ) @ ( times_3573771949741848930nteger @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1783_convex__bound__lt,axiom,
    ! [X: real,A: real,Y: real,U: real,V2: real] :
      ( ( ord_less_real @ X @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V2 )
           => ( ( ( plus_plus_real @ U @ V2 )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X ) @ ( times_times_real @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1784_convex__bound__lt,axiom,
    ! [X: rat,A: rat,Y: rat,U: rat,V2: rat] :
      ( ( ord_less_rat @ X @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V2 )
           => ( ( ( plus_plus_rat @ U @ V2 )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X ) @ ( times_times_rat @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1785_convex__bound__lt,axiom,
    ! [X: int,A: int,Y: int,U: int,V2: int] :
      ( ( ord_less_int @ X @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V2 )
           => ( ( ( plus_plus_int @ U @ V2 )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V2 @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_1786_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: list_VEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va3 @ Vb ) @ X )
      = ( ( X = Mi )
        | ( X = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1787_vebt__member_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) @ X3 ) )
       => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ X3 ) )
         => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_1788_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_1789_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_1790_geqmaxNone,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Ma @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = none_nat ) ) ) ).

% geqmaxNone
thf(fact_1791_vebt__pred_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,Uw: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ Uw ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,Va: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ Va ) ) ) )
         => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT,Vb2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va2 ) @ Vb2 ) )
           => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT,Vf: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Vf ) )
             => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT,Vj: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Vj ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% vebt_pred.cases
thf(fact_1792_vebt__succ_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ B3 ) @ zero_zero_nat ) )
     => ( ! [Uv2: $o,Uw: $o,N3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) ) )
       => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT,Va2: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Va2 ) )
         => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT,Ve: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Ve ) )
           => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT,Vi: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Vi ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ).

% vebt_succ.cases
thf(fact_1793_vebt__delete_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ zero_zero_nat ) )
     => ( ! [A3: $o,B3: $o] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ zero_zero_nat ) ) )
       => ( ! [A3: $o,B3: $o,N3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) )
         => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,Uu2: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Uu2 ) )
           => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ X3 ) )
             => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT,X3: nat] :
                    ( X
                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ X3 ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                      ( X
                     != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ) ) ).

% vebt_delete.cases
thf(fact_1794_vebt__insert_Ocases,axiom,
    ! [X: produc9072475918466114483BT_nat] :
      ( ! [A3: $o,B3: $o,X3: nat] :
          ( X
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ X3 ) )
     => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
            ( X
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ X3 ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
              ( X
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ X3 ) )
         => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                ( X
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) @ X3 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
                  ( X
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ X3 ) ) ) ) ) ) ).

% vebt_insert.cases
thf(fact_1795_succ__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_correct
thf(fact_1796_pred__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% pred_correct
thf(fact_1797_pred__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Px: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Px ) )
        = ( vEBT_is_pred_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Px ) ) ) ).

% pred_corr
thf(fact_1798_succ__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Sx: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Sx ) )
        = ( vEBT_is_succ_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X @ Sx ) ) ) ).

% succ_corr
thf(fact_1799_sum__squares__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1800_sum__squares__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1801_sum__squares__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_1802_vebt__succ_Osimps_I2_J,axiom,
    ! [Uv: $o,Uw2: $o,N: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uv @ Uw2 ) @ ( suc @ N ) )
      = none_nat ) ).

% vebt_succ.simps(2)
thf(fact_1803_vebt__pred_Osimps_I1_J,axiom,
    ! [Uu: $o,Uv: $o] :
      ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ Uu @ Uv ) @ zero_zero_nat )
      = none_nat ) ).

% vebt_pred.simps(1)
thf(fact_1804_vebt__succ_Osimps_I4_J,axiom,
    ! [V2: product_prod_nat_nat,Vc: list_VEBT_VEBT,Vd2: vEBT_VEBT,Ve2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vc @ Vd2 ) @ Ve2 )
      = none_nat ) ).

% vebt_succ.simps(4)
thf(fact_1805_vebt__pred_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vd2: list_VEBT_VEBT,Ve2: vEBT_VEBT,Vf2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Vd2 @ Ve2 ) @ Vf2 )
      = none_nat ) ).

% vebt_pred.simps(5)
thf(fact_1806_vebt__pred_Osimps_I6_J,axiom,
    ! [V2: product_prod_nat_nat,Vh2: list_VEBT_VEBT,Vi2: vEBT_VEBT,Vj2: nat] :
      ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vh2 @ Vi2 ) @ Vj2 )
      = none_nat ) ).

% vebt_pred.simps(6)
thf(fact_1807_vebt__succ_Osimps_I5_J,axiom,
    ! [V2: product_prod_nat_nat,Vg2: list_VEBT_VEBT,Vh2: vEBT_VEBT,Vi2: nat] :
      ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vg2 @ Vh2 ) @ Vi2 )
      = none_nat ) ).

% vebt_succ.simps(5)
thf(fact_1808_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_1809_vebt__succ_Osimps_I1_J,axiom,
    ! [B: $o,Uu: $o] :
      ( ( B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( vEBT_vebt_succ @ ( vEBT_Leaf @ Uu @ B ) @ zero_zero_nat )
          = none_nat ) ) ) ).

% vebt_succ.simps(1)
thf(fact_1810_vebt__pred_Osimps_I3_J,axiom,
    ! [B: $o,A: $o,Va3: nat] :
      ( ( B
       => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_pred @ ( vEBT_Leaf @ A @ B ) @ ( suc @ ( suc @ Va3 ) ) )
              = none_nat ) ) ) ) ) ).

% vebt_pred.simps(3)
thf(fact_1811_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_1812_vebt__delete_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( vEBT_vebt_delete @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat )
      = ( vEBT_Leaf @ $false @ B ) ) ).

% vebt_delete.simps(1)
thf(fact_1813_is__pred__in__set__def,axiom,
    ( vEBT_is_pred_in_set
    = ( ^ [Xs: set_nat,X4: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs )
          & ( ord_less_nat @ Y4 @ X4 )
          & ! [Z4: nat] :
              ( ( member_nat @ Z4 @ Xs )
             => ( ( ord_less_nat @ Z4 @ X4 )
               => ( ord_less_eq_nat @ Z4 @ Y4 ) ) ) ) ) ) ).

% is_pred_in_set_def
thf(fact_1814_is__succ__in__set__def,axiom,
    ( vEBT_is_succ_in_set
    = ( ^ [Xs: set_nat,X4: nat,Y4: nat] :
          ( ( member_nat @ Y4 @ Xs )
          & ( ord_less_nat @ X4 @ Y4 )
          & ! [Z4: nat] :
              ( ( member_nat @ Z4 @ Xs )
             => ( ( ord_less_nat @ X4 @ Z4 )
               => ( ord_less_eq_nat @ Y4 @ Z4 ) ) ) ) ) ) ).

% is_succ_in_set_def
thf(fact_1815_sum__squares__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1816_sum__squares__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1817_sum__squares__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_1818_sum__squares__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1819_sum__squares__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1820_sum__squares__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_1821_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_1822_vebt__delete_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,TrLst2: list_VEBT_VEBT,Smry2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ TrLst2 @ Smry2 ) ) ).

% vebt_delete.simps(5)
thf(fact_1823_vebt__delete_Osimps_I6_J,axiom,
    ! [Mi: nat,Ma: nat,Tr2: list_VEBT_VEBT,Sm2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ zero_zero_nat ) @ Tr2 @ Sm2 ) ) ).

% vebt_delete.simps(6)
thf(fact_1824_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1825_delete__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct
thf(fact_1826_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_1827_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_1828_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_1829_add__shift,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( plus_plus_nat @ X @ Y )
        = Z3 )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z3 ) ) ) ).

% add_shift
thf(fact_1830_delete__correct_H,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_delete @ T @ X ) )
        = ( minus_minus_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% delete_correct'
thf(fact_1831_obtain__set__succ,axiom,
    ! [X: nat,Z3: nat,A4: set_nat,B5: set_nat] :
      ( ( ord_less_nat @ X @ Z3 )
     => ( ( vEBT_VEBT_max_in_set @ A4 @ Z3 )
       => ( ( finite_finite_nat @ B5 )
         => ( ( A4 = B5 )
           => ? [X_12: nat] : ( vEBT_is_succ_in_set @ A4 @ X @ X_12 ) ) ) ) ) ).

% obtain_set_succ
thf(fact_1832_obtain__set__pred,axiom,
    ! [Z3: nat,X: nat,A4: set_nat] :
      ( ( ord_less_nat @ Z3 @ X )
     => ( ( vEBT_VEBT_min_in_set @ A4 @ Z3 )
       => ( ( finite_finite_nat @ A4 )
         => ? [X_12: nat] : ( vEBT_is_pred_in_set @ A4 @ X @ X_12 ) ) ) ) ).

% obtain_set_pred
thf(fact_1833_length__pos__if__in__set,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1834_length__pos__if__in__set,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1835_length__pos__if__in__set,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1836_length__pos__if__in__set,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1837_length__pos__if__in__set,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1838_length__pos__if__in__set,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1839_length__pos__if__in__set,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_pos_if_in_set
thf(fact_1840_option_Osize__gen_I2_J,axiom,
    ! [X: product_prod_nat_nat > nat,X2: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X @ ( some_P7363390416028606310at_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1841_option_Osize__gen_I2_J,axiom,
    ! [X: nat > nat,X2: nat] :
      ( ( size_option_nat @ X @ ( some_nat @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1842_option_Osize__gen_I2_J,axiom,
    ! [X: num > nat,X2: num] :
      ( ( size_option_num @ X @ ( some_num @ X2 ) )
      = ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1843_add__scale__eq__noteq,axiom,
    ! [R3: complex,A: complex,B: complex,C2: complex,D: complex] :
      ( ( R3 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R3 @ C2 ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1844_add__scale__eq__noteq,axiom,
    ! [R3: real,A: real,B: real,C2: real,D: real] :
      ( ( R3 != zero_zero_real )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R3 @ C2 ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1845_add__scale__eq__noteq,axiom,
    ! [R3: rat,A: rat,B: rat,C2: rat,D: rat] :
      ( ( R3 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R3 @ C2 ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1846_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C2: nat,D: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C2 ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1847_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C2: int,D: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C2 != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C2 ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_1848_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_1849_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 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ X5 @ A ) ) ) ) ).

% pred_none_empty
thf(fact_1850_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 )
       => ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ Xs2 )
              & ( ord_less_nat @ A @ X5 ) ) ) ) ).

% succ_none_empty
thf(fact_1851_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_1852_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_1853_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_1854_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_1855_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_1856_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_1857_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_1858_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_1859_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1860_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_1861_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_1862_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_1863_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1864_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_1865_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1866_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_1867_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_1868_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_1869_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_1870_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_1871_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_1872_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_1873_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_1874_add__diff__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1875_add__diff__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1876_add__diff__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1877_add__diff__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_1878_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_1879_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_1880_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_1881_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_1882_add__diff__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C2 @ A ) @ ( plus_plus_real @ C2 @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1883_add__diff__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C2 @ A ) @ ( plus_plus_rat @ C2 @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1884_add__diff__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1885_add__diff__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_1886_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1887_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1888_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_1889_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1890_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1891_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_1892_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_1893_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_1894_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_1895_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_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_diff__numeral__special_I9_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(9)
thf(fact_1908_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_1909_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_1910_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_1911_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_1912_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_1913_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_1914_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_1915_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_1916_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1917_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1918_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1919_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_1920_diff__right__commute,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 ) ) ).

% diff_right_commute
thf(fact_1921_diff__right__commute,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).

% diff_right_commute
thf(fact_1922_diff__right__commute,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).

% diff_right_commute
thf(fact_1923_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D ) )
     => ( ( A = B )
        = ( C2 = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1924_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( A = B )
        = ( C2 = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_1925_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C2 )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1926_diff__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C2 )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_1927_diff__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_1928_diff__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_left_mono
thf(fact_1929_diff__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1930_diff__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_right_mono
thf(fact_1931_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C2 @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1932_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C2 @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_1933_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: complex,Z2: complex] : Y5 = Z2 )
    = ( ^ [A5: complex,B4: complex] :
          ( ( minus_minus_complex @ A5 @ B4 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1934_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: real,Z2: real] : Y5 = Z2 )
    = ( ^ [A5: real,B4: real] :
          ( ( minus_minus_real @ A5 @ B4 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1935_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: rat,Z2: rat] : Y5 = Z2 )
    = ( ^ [A5: rat,B4: rat] :
          ( ( minus_minus_rat @ A5 @ B4 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1936_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y5: int,Z2: int] : Y5 = Z2 )
    = ( ^ [A5: int,B4: int] :
          ( ( minus_minus_int @ A5 @ B4 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_1937_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1938_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1939_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).

% diff_strict_right_mono
thf(fact_1940_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C2 @ A ) @ ( minus_minus_real @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1941_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C2 @ A ) @ ( minus_minus_rat @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1942_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_1943_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C2 @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1944_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C2 @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1945_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C2 @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C2 @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_1946_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C2 )
       => ( ord_less_real @ ( minus_minus_real @ A @ C2 ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1947_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C2 )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C2 ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1948_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C2: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C2 )
       => ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_1949_left__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1950_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1951_left__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) ) ) ).

% left_diff_distrib
thf(fact_1952_right__diff__distrib,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1953_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1954_right__diff__distrib,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% right_diff_distrib
thf(fact_1955_left__diff__distrib_H,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C2 ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1956_left__diff__distrib_H,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C2 ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1957_left__diff__distrib_H,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C2 ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1958_left__diff__distrib_H,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C2 ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) ) ) ).

% left_diff_distrib'
thf(fact_1959_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1960_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1961_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1962_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) ) ) ).

% right_diff_distrib'
thf(fact_1963_diff__diff__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_1964_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_1965_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_1966_diff__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).

% diff_diff_eq
thf(fact_1967_add__implies__diff,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1968_add__implies__diff,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1969_add__implies__diff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1970_add__implies__diff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C2 @ B )
        = A )
     => ( C2
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_1971_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1972_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1973_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_1974_diff__add__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_1975_diff__add__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_1976_diff__add__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_add_eq
thf(fact_1977_diff__diff__eq2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1978_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1979_diff__diff__eq2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).

% diff_diff_eq2
thf(fact_1980_add__diff__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_1981_add__diff__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_1982_add__diff__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% add_diff_eq
thf(fact_1983_eq__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C2 @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_1984_eq__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C2 @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_1985_eq__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C2 @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C2 ) ) ).

% eq_diff_eq
thf(fact_1986_diff__eq__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_1987_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_1988_diff__eq__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C2 )
      = ( A
        = ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_eq_eq
thf(fact_1989_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_1990_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_1991_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_1992_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_nat @ X4 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1993_bounded__nat__set__is__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ N6 )
         => ( ord_less_nat @ X3 @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1994_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M6: nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ N5 )
         => ( ord_less_eq_nat @ X4 @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1995_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A5: real,B4: real] : ( ord_less_eq_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_1996_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_1997_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_1998_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A5: real,B4: real] : ( ord_less_real @ ( minus_minus_real @ A5 @ B4 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_1999_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A5: rat,B4: rat] : ( ord_less_rat @ ( minus_minus_rat @ A5 @ B4 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_2000_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A5: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A5 @ B4 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_2001_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K2: real,N: real,J2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
     => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K2 ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
         => ( ( ord_less_eq_real @ N @ ( plus_plus_real @ J2 @ K2 ) )
           => ( ord_less_eq_real @ ( minus_minus_real @ N @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2002_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K2: rat,N: rat,J2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
     => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J2 @ K2 ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
         => ( ( ord_less_eq_rat @ N @ ( plus_plus_rat @ J2 @ K2 ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2003_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K2: nat,N: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
     => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K2 ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
         => ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J2 @ K2 ) )
           => ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2004_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K2: int,N: int,J2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
     => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K2 ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
         => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J2 @ K2 ) )
           => ( ord_less_eq_int @ ( minus_minus_int @ N @ K2 ) @ J2 ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_2005_add__le__imp__le__diff,axiom,
    ! [I2: real,K2: real,N: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K2 ) @ N )
     => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2006_add__le__imp__le__diff,axiom,
    ! [I2: rat,K2: rat,N: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K2 ) @ N )
     => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2007_add__le__imp__le__diff,axiom,
    ! [I2: nat,K2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ N )
     => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2008_add__le__imp__le__diff,axiom,
    ! [I2: int,K2: int,N: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K2 ) @ N )
     => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K2 ) ) ) ).

% add_le_imp_le_diff
thf(fact_2009_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C2 )
          = ( B
            = ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_2010_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_2011_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_2012_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_2013_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_2014_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
        = ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_2015_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_2016_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_2017_le__add__diff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).

% le_add_diff
thf(fact_2018_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_2019_le__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C2 @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2020_le__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2021_le__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% le_diff_eq
thf(fact_2022_diff__le__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2023_diff__le__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2024_diff__le__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_le_eq
thf(fact_2025_less__diff__eq,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C2 @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2026_less__diff__eq,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C2 @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2027_less__diff__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).

% less_diff_eq
thf(fact_2028_diff__less__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C2 )
      = ( ord_less_real @ A @ ( plus_plus_real @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2029_diff__less__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C2 )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2030_diff__less__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
      = ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).

% diff_less_eq
thf(fact_2031_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_2032_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_2033_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_2034_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_2035_square__diff__square__factored,axiom,
    ! [X: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2036_square__diff__square__factored,axiom,
    ! [X: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_rat @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2037_square__diff__square__factored,axiom,
    ! [X: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_2038_eq__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( C2
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2039_eq__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( C2
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2040_eq__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( C2
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_2041_eq__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 )
        = ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 )
        = D ) ) ).

% eq_add_iff1
thf(fact_2042_eq__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C2 )
        = D ) ) ).

% eq_add_iff1
thf(fact_2043_eq__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 )
        = ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 )
        = D ) ) ).

% eq_add_iff1
thf(fact_2044_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_eq_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2045_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_eq_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2046_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_eq_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_2047_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( 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 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2048_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( 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 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2049_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( 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 ) @ C2 ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_2050_less__add__iff1,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_2051_less__add__iff1,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_2052_less__add__iff1,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C2 ) @ D ) ) ).

% less_add_iff1
thf(fact_2053_less__add__iff2,axiom,
    ! [A: real,E: real,C2: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E ) @ C2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E ) @ D ) )
      = ( ord_less_real @ C2 @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2054_less__add__iff2,axiom,
    ! [A: rat,E: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E ) @ C2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E ) @ D ) )
      = ( ord_less_rat @ C2 @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2055_less__add__iff2,axiom,
    ! [A: int,E: int,C2: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
      = ( ord_less_int @ C2 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).

% less_add_iff2
thf(fact_2056_square__diff__one__factored,axiom,
    ! [X: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( times_3573771949741848930nteger @ X @ X ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ X @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) ) ) ).

% square_diff_one_factored
thf(fact_2057_square__diff__one__factored,axiom,
    ! [X: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X @ X ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_2058_square__diff__one__factored,axiom,
    ! [X: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ X ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_2059_square__diff__one__factored,axiom,
    ! [X: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ X ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_2060_square__diff__one__factored,axiom,
    ! [X: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_2061_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_complex @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2062_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: set_nat > $o,F2: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2063_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: set_int > $o,F2: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2064_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F2: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_rat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_real @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2065_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F2: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_complex @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2066_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: set_nat > $o,F2: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2067_finite__ranking__induct,axiom,
    ! [S3: set_int,P2: set_int > $o,F2: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( P2 @ bot_bot_set_int )
       => ( ! [X3: int,S4: set_int] :
              ( ( finite_finite_int @ S4 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_int @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2068_finite__ranking__induct,axiom,
    ! [S3: set_real,P2: set_real > $o,F2: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P2 @ bot_bot_set_real )
       => ( ! [X3: real,S4: set_real] :
              ( ( finite_finite_real @ S4 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S4 )
                   => ( ord_less_eq_num @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_real @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2069_finite__ranking__induct,axiom,
    ! [S3: set_complex,P2: set_complex > $o,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P2 @ bot_bot_set_complex )
       => ( ! [X3: complex,S4: set_complex] :
              ( ( finite3207457112153483333omplex @ S4 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_complex @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2070_finite__ranking__induct,axiom,
    ! [S3: set_nat,P2: set_nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P2 @ bot_bot_set_nat )
       => ( ! [X3: nat,S4: set_nat] :
              ( ( finite_finite_nat @ S4 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S4 )
                   => ( ord_less_eq_nat @ ( F2 @ Y6 ) @ ( F2 @ X3 ) ) )
               => ( ( P2 @ S4 )
                 => ( P2 @ ( insert_nat @ X3 @ S4 ) ) ) ) )
         => ( P2 @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_2071_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs2 ) ) ).

% length_induct
thf(fact_2072_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs2 ) ) ).

% length_induct
thf(fact_2073_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs2 ) ) ).

% length_induct
thf(fact_2074_length__induct,axiom,
    ! [P2: 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 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs2 ) ) ).

% length_induct
thf(fact_2075_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_2076_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_2077_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_2078_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_2079_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_2080_crossproduct__eq,axiom,
    ! [W2: real,Y: real,X: real,Z3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y ) @ ( times_times_real @ X @ Z3 ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z3 ) @ ( times_times_real @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2081_crossproduct__eq,axiom,
    ! [W2: rat,Y: rat,X: rat,Z3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y ) @ ( times_times_rat @ X @ Z3 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z3 ) @ ( times_times_rat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2082_crossproduct__eq,axiom,
    ! [W2: nat,Y: nat,X: nat,Z3: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y ) @ ( times_times_nat @ X @ Z3 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z3 ) @ ( times_times_nat @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2083_crossproduct__eq,axiom,
    ! [W2: int,Y: int,X: int,Z3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y ) @ ( times_times_int @ X @ Z3 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z3 ) @ ( times_times_int @ X @ Y ) ) )
      = ( ( W2 = X )
        | ( Y = Z3 ) ) ) ).

% crossproduct_eq
thf(fact_2084_crossproduct__noteq,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ( A != B )
        & ( C2 != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2085_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ( A != B )
        & ( C2 != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2086_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ( A != B )
        & ( C2 != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2087_crossproduct__noteq,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ( A != B )
        & ( C2 != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C2 ) ) ) ) ).

% crossproduct_noteq
thf(fact_2088_option_Osize__gen_I1_J,axiom,
    ! [X: nat > nat] :
      ( ( size_option_nat @ X @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2089_option_Osize__gen_I1_J,axiom,
    ! [X: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2090_option_Osize__gen_I1_J,axiom,
    ! [X: num > nat] :
      ( ( size_option_num @ X @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_2091_finite__has__minimal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2092_finite__has__minimal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2093_finite__has__minimal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2094_finite__has__minimal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2095_finite__has__minimal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2096_finite__has__minimal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_2097_finite__has__maximal,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( A4 != bot_bot_set_real )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2098_finite__has__maximal,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( A4 != bot_bot_set_set_nat )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2099_finite__has__maximal,axiom,
    ! [A4: set_rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( A4 != bot_bot_set_rat )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2100_finite__has__maximal,axiom,
    ! [A4: set_num] :
      ( ( finite_finite_num @ A4 )
     => ( ( A4 != bot_bot_set_num )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2101_finite__has__maximal,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( A4 != bot_bot_set_nat )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2102_finite__has__maximal,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( A4 != bot_bot_set_int )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_2103_mult__diff__mult,axiom,
    ! [X: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2104_mult__diff__mult,axiom,
    ! [X: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2105_mult__diff__mult,axiom,
    ! [X: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_2106_diff__shunt__var,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( minus_minus_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2107_diff__shunt__var,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( minus_minus_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2108_diff__shunt__var,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2109_diff__shunt__var,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( minus_minus_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% diff_shunt_var
thf(fact_2110_minNullmin,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ T )
     => ( ( vEBT_vebt_mint @ T )
        = none_nat ) ) ).

% minNullmin
thf(fact_2111_minminNull,axiom,
    ! [T: vEBT_VEBT] :
      ( ( ( vEBT_vebt_mint @ T )
        = none_nat )
     => ( vEBT_VEBT_minNull @ T ) ) ).

% minminNull
thf(fact_2112_vebt__maxt_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( ( A3
                       => ( Y
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_2113_vebt__mint_Opelims,axiom,
    ! [X: vEBT_VEBT,Y: option_nat] :
      ( ( ( vEBT_vebt_mint @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( ( A3
                   => ( Y
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A3
                   => ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( Y = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A3 @ B3 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
               => ( ( Y = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux @ Uy @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_2114_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M6: nat] :
          ? [N4: nat] :
            ( ( ord_less_eq_nat @ M6 @ N4 )
            & ( member_nat @ N4 @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_2115_inthall,axiom,
    ! [Xs2: list_complex,P2: complex > $o,N: nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ ( set_complex2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( P2 @ ( nth_complex @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2116_inthall,axiom,
    ! [Xs2: list_real,P2: real > $o,N: nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_real2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( P2 @ ( nth_real @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2117_inthall,axiom,
    ! [Xs2: list_set_nat,P2: set_nat > $o,N: nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( P2 @ ( nth_set_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2118_inthall,axiom,
    ! [Xs2: list_VEBT_VEBT,P2: vEBT_VEBT > $o,N: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2119_inthall,axiom,
    ! [Xs2: list_o,P2: $o > $o,N: nat] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( P2 @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2120_inthall,axiom,
    ! [Xs2: list_nat,P2: nat > $o,N: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( P2 @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2121_inthall,axiom,
    ! [Xs2: list_int,P2: int > $o,N: nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( P2 @ X3 ) )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( P2 @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% inthall
thf(fact_2122_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_2123_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X ) ) ).

% min_Null_member
thf(fact_2124_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_2125_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_2126_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_2127_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_2128_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_2129_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K2 )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ).

% diff_diff_left
thf(fact_2130_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_2131_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_2132_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_2133_Nat_Oadd__diff__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 ) ) ) ).

% Nat.add_diff_assoc
thf(fact_2134_Nat_Oadd__diff__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K2 ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_2135_Nat_Odiff__diff__right,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_2136_diff__Suc__1,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
      = N ) ).

% diff_Suc_1
thf(fact_2137_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_2138_diff__Suc__diff__eq1,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_2139_diff__Suc__diff__eq2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K2 ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_2140_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_2141_zero__induct__lemma,axiom,
    ! [P2: nat > $o,K2: nat,I2: nat] :
      ( ( P2 @ K2 )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ ( minus_minus_nat @ K2 @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_2142_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_2143_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_2144_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_2145_less__imp__diff__less,axiom,
    ! [J2: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ J2 @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_2146_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_2147_le__diff__iff_H,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_2148_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_2156_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_2157_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_2158_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_2159_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_2160_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_2161_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_2162_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_VEBT_VEBT,Z2: list_VEBT_VEBT] : Y5 = Z2 )
    = ( ^ [Xs: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( nth_VEBT_VEBT @ Xs @ I )
                = ( nth_VEBT_VEBT @ Ys3 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2163_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_o,Z2: list_o] : Y5 = Z2 )
    = ( ^ [Xs: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
             => ( ( nth_o @ Xs @ I )
                = ( nth_o @ Ys3 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2164_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_nat,Z2: list_nat] : Y5 = Z2 )
    = ( ^ [Xs: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
             => ( ( nth_nat @ Xs @ I )
                = ( nth_nat @ Ys3 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2165_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y5: list_int,Z2: list_int] : Y5 = Z2 )
    = ( ^ [Xs: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
             => ( ( nth_int @ Xs @ I )
                = ( nth_int @ Ys3 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_2166_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > vEBT_VEBT > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: vEBT_VEBT] : ( P2 @ I @ X7 ) ) )
      = ( ? [Xs: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_VEBT_VEBT @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2167_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > $o > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: $o] : ( P2 @ I @ X7 ) ) )
      = ( ? [Xs: list_o] :
            ( ( ( size_size_list_o @ Xs )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_o @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2168_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: nat] : ( P2 @ I @ X7 ) ) )
      = ( ? [Xs: list_nat] :
            ( ( ( size_size_list_nat @ Xs )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_nat @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2169_Skolem__list__nth,axiom,
    ! [K2: nat,P2: nat > int > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: int] : ( P2 @ I @ X7 ) ) )
      = ( ? [Xs: list_int] :
            ( ( ( size_size_list_int @ Xs )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P2 @ I @ ( nth_int @ Xs @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_2170_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_2171_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_2172_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_2173_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_2174_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_2175_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_2176_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_2177_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_2178_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_2179_diff__less__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_2180_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_2181_finite__maxlen,axiom,
    ! [M5: set_list_VEBT_VEBT] :
      ( ( finite3004134309566078307T_VEBT @ M5 )
     => ? [N3: nat] :
        ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ M5 )
         => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X5 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2182_finite__maxlen,axiom,
    ! [M5: set_list_o] :
      ( ( finite_finite_list_o @ M5 )
     => ? [N3: nat] :
        ! [X5: list_o] :
          ( ( member_list_o @ X5 @ M5 )
         => ( ord_less_nat @ ( size_size_list_o @ X5 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2183_finite__maxlen,axiom,
    ! [M5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ M5 )
     => ? [N3: nat] :
        ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ M5 )
         => ( ord_less_nat @ ( size_size_list_nat @ X5 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2184_finite__maxlen,axiom,
    ! [M5: set_list_int] :
      ( ( finite3922522038869484883st_int @ M5 )
     => ? [N3: nat] :
        ! [X5: list_int] :
          ( ( member_list_int @ X5 @ M5 )
         => ( ord_less_nat @ ( size_size_list_int @ X5 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_2185_less__diff__conv,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ).

% less_diff_conv
thf(fact_2186_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_2187_le__diff__conv,axiom,
    ! [J2: nat,K2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
      = ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I2 @ K2 ) ) ) ).

% le_diff_conv
thf(fact_2188_Nat_Ole__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J2 ) ) ) ).

% Nat.le_diff_conv2
thf(fact_2189_Nat_Odiff__add__assoc,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K2 )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K2 ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_2190_Nat_Odiff__add__assoc2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K2 )
        = ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_2191_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ( minus_minus_nat @ J2 @ I2 )
          = K2 )
        = ( J2
          = ( plus_plus_nat @ K2 @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_2192_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_2193_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va3: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va3 @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_2194_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_2195_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_2196_nth__mem,axiom,
    ! [N: nat,Xs2: list_set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ ( nth_set_nat @ Xs2 @ N ) @ ( set_set_nat2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_2197_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_2198_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_2199_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_2200_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_2201_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ! [X3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2202_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_o,P2: $o > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ! [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_o @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2203_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_nat,P2: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_nat @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2204_list__ball__nth,axiom,
    ! [N: nat,Xs2: list_int,P2: int > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_int @ Xs2 @ N ) ) ) ) ).

% list_ball_nth
thf(fact_2205_in__set__conv__nth,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2206_in__set__conv__nth,axiom,
    ! [X: real,Xs2: list_real] :
      ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2207_in__set__conv__nth,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2208_in__set__conv__nth,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2209_in__set__conv__nth,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2210_in__set__conv__nth,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2211_in__set__conv__nth,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_2212_all__nth__imp__all__set,axiom,
    ! [Xs2: list_complex,P2: complex > $o,X: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
         => ( P2 @ ( nth_complex @ Xs2 @ I3 ) ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2213_all__nth__imp__all__set,axiom,
    ! [Xs2: list_real,P2: real > $o,X: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs2 ) )
         => ( P2 @ ( nth_real @ Xs2 @ I3 ) ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2214_all__nth__imp__all__set,axiom,
    ! [Xs2: list_set_nat,P2: set_nat > $o,X: set_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
         => ( P2 @ ( nth_set_nat @ Xs2 @ I3 ) ) )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2215_all__nth__imp__all__set,axiom,
    ! [Xs2: list_VEBT_VEBT,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2216_all__nth__imp__all__set,axiom,
    ! [Xs2: list_o,P2: $o > $o,X: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs2 ) )
         => ( P2 @ ( nth_o @ Xs2 @ I3 ) ) )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2217_all__nth__imp__all__set,axiom,
    ! [Xs2: list_nat,P2: nat > $o,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs2 ) )
         => ( P2 @ ( nth_nat @ Xs2 @ I3 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2218_all__nth__imp__all__set,axiom,
    ! [Xs2: list_int,P2: int > $o,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs2 ) )
         => ( P2 @ ( nth_int @ Xs2 @ I3 ) ) )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_2219_all__set__conv__all__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2220_all__set__conv__all__nth,axiom,
    ! [Xs2: list_o,P2: $o > $o] :
      ( ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
           => ( P2 @ ( nth_o @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2221_all__set__conv__all__nth,axiom,
    ! [Xs2: list_nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
           => ( P2 @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2222_all__set__conv__all__nth,axiom,
    ! [Xs2: list_int,P2: int > $o] :
      ( ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
           => ( P2 @ ( nth_int @ Xs2 @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_2223_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_2224_nat__diff__split,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P2 @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P2 @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_2225_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P2 @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_2226_less__diff__conv2,axiom,
    ! [K2: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K2 @ J2 )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J2 @ K2 ) @ I2 )
        = ( ord_less_nat @ J2 @ ( plus_plus_nat @ I2 @ K2 ) ) ) ) ).

% less_diff_conv2
thf(fact_2227_nat__eq__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 )
          = N ) ) ) ).

% nat_eq_add_iff1
thf(fact_2228_nat__eq__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 )
          = ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( M2
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_2229_nat__le__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_le_add_iff1
thf(fact_2230_nat__le__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_le_add_iff2
thf(fact_2231_nat__diff__add__eq1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_diff_add_eq1
thf(fact_2232_nat__diff__add__eq2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( minus_minus_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_2233_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ! [Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2234_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( X
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2235_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_2236_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_2237_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% add_eq_if
thf(fact_2238_nat__less__add__iff1,axiom,
    ! [J2: nat,I2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).

% nat_less_add_iff1
thf(fact_2239_nat__less__add__iff2,axiom,
    ! [I2: nat,J2: nat,U: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M2 ) @ ( plus_plus_nat @ ( times_times_nat @ J2 @ U ) @ N ) )
        = ( ord_less_nat @ M2 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J2 @ I2 ) @ U ) @ N ) ) ) ) ).

% nat_less_add_iff2
thf(fact_2240_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N4 ) ) ) ) ) ).

% mult_eq_if
thf(fact_2241_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( ( X
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y )
       => ( ( ? [Uv2: $o] :
                ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y )
         => ( ( ? [Uu2: $o] :
                  ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y )
           => ( ( ? [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ~ Y )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va2 @ Vb2 @ Vc2 ) )
                 => Y ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2242_field__lbound__gt__zero,axiom,
    ! [D1: real,D22: real] :
      ( ( ord_less_real @ zero_zero_real @ D1 )
     => ( ( ord_less_real @ zero_zero_real @ D22 )
       => ? [E2: real] :
            ( ( ord_less_real @ zero_zero_real @ E2 )
            & ( ord_less_real @ E2 @ D1 )
            & ( ord_less_real @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2243_field__lbound__gt__zero,axiom,
    ! [D1: rat,D22: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D22 )
       => ? [E2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E2 )
            & ( ord_less_rat @ E2 @ D1 )
            & ( ord_less_rat @ E2 @ D22 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_2244_finite__has__minimal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ X3 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2245_finite__has__minimal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ X3 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2246_finite__has__minimal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ X3 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2247_finite__has__minimal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ X3 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2248_finite__has__minimal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ X3 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2249_finite__has__minimal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ X3 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ Xa @ X3 )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_2250_finite__has__maximal2,axiom,
    ! [A4: set_real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ A @ A4 )
       => ? [X3: real] :
            ( ( member_real @ X3 @ A4 )
            & ( ord_less_eq_real @ A @ X3 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A4 )
               => ( ( ord_less_eq_real @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2251_finite__has__maximal2,axiom,
    ! [A4: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ A @ A4 )
       => ? [X3: set_nat] :
            ( ( member_set_nat @ X3 @ A4 )
            & ( ord_less_eq_set_nat @ A @ X3 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A4 )
               => ( ( ord_less_eq_set_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2252_finite__has__maximal2,axiom,
    ! [A4: set_rat,A: rat] :
      ( ( finite_finite_rat @ A4 )
     => ( ( member_rat @ A @ A4 )
       => ? [X3: rat] :
            ( ( member_rat @ X3 @ A4 )
            & ( ord_less_eq_rat @ A @ X3 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A4 )
               => ( ( ord_less_eq_rat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2253_finite__has__maximal2,axiom,
    ! [A4: set_num,A: num] :
      ( ( finite_finite_num @ A4 )
     => ( ( member_num @ A @ A4 )
       => ? [X3: num] :
            ( ( member_num @ X3 @ A4 )
            & ( ord_less_eq_num @ A @ X3 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A4 )
               => ( ( ord_less_eq_num @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2254_finite__has__maximal2,axiom,
    ! [A4: set_nat,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ A @ A4 )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
            & ( ord_less_eq_nat @ A @ X3 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A4 )
               => ( ( ord_less_eq_nat @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2255_finite__has__maximal2,axiom,
    ! [A4: set_int,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ A @ A4 )
       => ? [X3: int] :
            ( ( member_int @ X3 @ A4 )
            & ( ord_less_eq_int @ A @ X3 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A4 )
               => ( ( ord_less_eq_int @ X3 @ Xa )
                 => ( X3 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_2256_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Y: $o] :
      ( ( ( vEBT_VEBT_minNull @ X )
        = Y )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
                   => ( Y
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X
                        = ( 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_2257_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ( ( X
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_2258_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
       => ( ! [Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va2: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X
                    = ( 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_2259_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs2 ) @ M2 )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N @ M2 ) @ ( nth_VEBT_VEBT @ Xs2 @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2260_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs2: list_o,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N @ Xs2 ) @ M2 )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N @ M2 ) @ ( nth_o @ Xs2 @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2261_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs2: list_nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs2 ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs2 @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2262_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs2: list_int,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N @ Xs2 ) @ M2 )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N @ M2 ) @ ( nth_int @ Xs2 @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_2263_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F2 @ ( lattic4729654577720512673ex_rat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2264_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F2 @ ( lattic6811802900495863747at_rat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2265_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F2: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F2 @ ( lattic7811156612396918303nt_rat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2266_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F2: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_rat @ ( F2 @ ( lattic4420706379359479199al_rat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2267_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_num @ ( F2 @ ( lattic1922116423962787043ex_num @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2268_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_num @ ( F2 @ ( lattic4004264746738138117at_num @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2269_arg__min__least,axiom,
    ! [S3: set_int,Y: int,F2: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y @ S3 )
         => ( ord_less_eq_num @ ( F2 @ ( lattic5003618458639192673nt_num @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2270_arg__min__least,axiom,
    ! [S3: set_real,Y: real,F2: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y @ S3 )
         => ( ord_less_eq_num @ ( F2 @ ( lattic1613168225601753569al_num @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2271_arg__min__least,axiom,
    ! [S3: set_complex,Y: complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic5364784637807008409ex_nat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2272_arg__min__least,axiom,
    ! [S3: set_nat,Y: nat,F2: nat > nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y @ S3 )
         => ( ord_less_eq_nat @ ( F2 @ ( lattic7446932960582359483at_nat @ F2 @ S3 ) ) @ ( F2 @ Y ) ) ) ) ) ).

% arg_min_least
thf(fact_2273_Euclid__induct,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P2 @ A3 @ B3 )
          = ( P2 @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P2 @ A3 @ B3 )
             => ( P2 @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_2274_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M2: nat] :
      ( ! [K: nat] :
          ( ( ord_less_nat @ N @ K )
         => ( P2 @ K ) )
     => ( ! [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K ) ) )
       => ( P2 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_2275_length__mul__elem,axiom,
    ! [Xs2: list_list_VEBT_VEBT,N: nat] :
      ( ! [X3: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X3 @ ( set_list_VEBT_VEBT2 @ Xs2 ) )
         => ( ( size_s6755466524823107622T_VEBT @ X3 )
            = N ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs2 ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2276_length__mul__elem,axiom,
    ! [Xs2: list_list_o,N: nat] :
      ( ! [X3: list_o] :
          ( ( member_list_o @ X3 @ ( set_list_o2 @ Xs2 ) )
         => ( ( size_size_list_o @ X3 )
            = N ) )
     => ( ( size_size_list_o @ ( concat_o @ Xs2 ) )
        = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2277_length__mul__elem,axiom,
    ! [Xs2: list_list_nat,N: nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs2 ) )
         => ( ( size_size_list_nat @ X3 )
            = N ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs2 ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2278_length__mul__elem,axiom,
    ! [Xs2: list_list_int,N: nat] :
      ( ! [X3: list_int] :
          ( ( member_list_int @ X3 @ ( set_list_int2 @ Xs2 ) )
         => ( ( size_size_list_int @ X3 )
            = N ) )
     => ( ( size_size_list_int @ ( concat_int @ Xs2 ) )
        = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs2 ) @ N ) ) ) ).

% length_mul_elem
thf(fact_2279_triangle__Suc,axiom,
    ! [N: nat] :
      ( ( nat_triangle @ ( suc @ N ) )
      = ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).

% triangle_Suc
thf(fact_2280_length__enumerate,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_enumerate
thf(fact_2281_length__enumerate,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_enumerate
thf(fact_2282_length__enumerate,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_enumerate
thf(fact_2283_length__enumerate,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_enumerate
thf(fact_2284_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_2285_diff__commute,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J2 ) ) ).

% diff_commute
thf(fact_2286_prod__decode__aux_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_2287_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_2288_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_2289_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_2290_in__set__product__lists__length,axiom,
    ! [Xs2: list_VEBT_VEBT,Xss: list_list_VEBT_VEBT] :
      ( ( member2936631157270082147T_VEBT @ Xs2 @ ( set_list_VEBT_VEBT2 @ ( produc3021084454716106787T_VEBT @ Xss ) ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s8217280938318005548T_VEBT @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_2291_in__set__product__lists__length,axiom,
    ! [Xs2: list_o,Xss: list_list_o] :
      ( ( member_list_o @ Xs2 @ ( set_list_o2 @ ( product_lists_o @ Xss ) ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_s2710708370519433104list_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_2292_in__set__product__lists__length,axiom,
    ! [Xs2: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_2293_in__set__product__lists__length,axiom,
    ! [Xs2: list_int,Xss: list_list_int] :
      ( ( member_list_int @ Xs2 @ ( set_list_int2 @ ( product_lists_int @ Xss ) ) )
     => ( ( size_size_list_int @ Xs2 )
        = ( size_s533118279054570080st_int @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_2294_Gcd__remove0__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( gcd_Gcd_nat @ M5 )
        = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M5 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).

% Gcd_remove0_nat
thf(fact_2295_inf__period_I2_J,axiom,
    ! [P2: real > $o,D3: real,Q: real > $o] :
      ( ! [X3: real,K: real] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D3 ) ) ) )
     => ( ! [X3: real,K: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D3 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P2 @ X5 )
              | ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2296_inf__period_I2_J,axiom,
    ! [P2: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P2 @ X5 )
              | ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2297_inf__period_I2_J,axiom,
    ! [P2: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P2 @ X5 )
              | ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
              | ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(2)
thf(fact_2298_inf__period_I1_J,axiom,
    ! [P2: real > $o,D3: real,Q: real > $o] :
      ( ! [X3: real,K: real] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D3 ) ) ) )
     => ( ! [X3: real,K: real] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_real @ X3 @ ( times_times_real @ K @ D3 ) ) ) )
       => ! [X5: real,K4: real] :
            ( ( ( P2 @ X5 )
              & ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2299_inf__period_I1_J,axiom,
    ! [P2: rat > $o,D3: rat,Q: rat > $o] :
      ( ! [X3: rat,K: rat] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
     => ( ! [X3: rat,K: rat] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_rat @ X3 @ ( times_times_rat @ K @ D3 ) ) ) )
       => ! [X5: rat,K4: rat] :
            ( ( ( P2 @ X5 )
              & ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2300_inf__period_I1_J,axiom,
    ! [P2: int > $o,D3: int,Q: int > $o] :
      ( ! [X3: int,K: int] :
          ( ( P2 @ X3 )
          = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
     => ( ! [X3: int,K: int] :
            ( ( Q @ X3 )
            = ( Q @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
       => ! [X5: int,K4: int] :
            ( ( ( P2 @ X5 )
              & ( Q @ X5 ) )
            = ( ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) )
              & ( Q @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) ) ) ) ) ) ).

% inf_period(1)
thf(fact_2301_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_2302_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_2303_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_2304_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_2305_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_2306_Gcd__empty,axiom,
    ( ( gcd_Gcd_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_empty
thf(fact_2307_Gcd__empty,axiom,
    ( ( gcd_Gcd_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_empty
thf(fact_2308_Gcd__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( gcd_Gcd_int @ A4 )
        = zero_zero_int )
      = ( ord_less_eq_set_int @ A4 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) ) ) ).

% Gcd_0_iff
thf(fact_2309_Gcd__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( gcd_Gcd_nat @ A4 )
        = zero_zero_nat )
      = ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).

% Gcd_0_iff
thf(fact_2310_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_2311_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_2312_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_2313_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_2314_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2315_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2316_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2317_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2318_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_2319_prod__decode__def,axiom,
    ( nat_prod_decode
    = ( nat_prod_decode_aux @ zero_zero_nat ) ) ).

% prod_decode_def
thf(fact_2320_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_2321_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_2322_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_2323_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_2324_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_2325_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z @ X5 )
     => ~ ( ord_less_eq_real @ X5 @ T ) ) ).

% pinf(6)
thf(fact_2326_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z @ X5 )
     => ~ ( ord_less_eq_rat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_2327_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z @ X5 )
     => ~ ( ord_less_eq_num @ X5 @ T ) ) ).

% pinf(6)
thf(fact_2328_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z @ X5 )
     => ~ ( ord_less_eq_nat @ X5 @ T ) ) ).

% pinf(6)
thf(fact_2329_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z @ X5 )
     => ~ ( ord_less_eq_int @ X5 @ T ) ) ).

% pinf(6)
thf(fact_2330_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( ( ord_less_real @ Z @ X5 )
     => ( ord_less_eq_real @ T @ X5 ) ) ).

% pinf(8)
thf(fact_2331_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ Z @ X5 )
     => ( ord_less_eq_rat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_2332_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( ( ord_less_num @ Z @ X5 )
     => ( ord_less_eq_num @ T @ X5 ) ) ).

% pinf(8)
thf(fact_2333_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ Z @ X5 )
     => ( ord_less_eq_nat @ T @ X5 ) ) ).

% pinf(8)
thf(fact_2334_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( ( ord_less_int @ Z @ X5 )
     => ( ord_less_eq_int @ T @ X5 ) ) ).

% pinf(8)
thf(fact_2335_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z )
     => ( ord_less_eq_real @ X5 @ T ) ) ).

% minf(6)
thf(fact_2336_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z )
     => ( ord_less_eq_rat @ X5 @ T ) ) ).

% minf(6)
thf(fact_2337_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z )
     => ( ord_less_eq_num @ X5 @ T ) ) ).

% minf(6)
thf(fact_2338_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z )
     => ( ord_less_eq_nat @ X5 @ T ) ) ).

% minf(6)
thf(fact_2339_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z )
     => ( ord_less_eq_int @ X5 @ T ) ) ).

% minf(6)
thf(fact_2340_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z: real] :
    ! [X5: real] :
      ( ( ord_less_real @ X5 @ Z )
     => ~ ( ord_less_eq_real @ T @ X5 ) ) ).

% minf(8)
thf(fact_2341_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z: rat] :
    ! [X5: rat] :
      ( ( ord_less_rat @ X5 @ Z )
     => ~ ( ord_less_eq_rat @ T @ X5 ) ) ).

% minf(8)
thf(fact_2342_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z: num] :
    ! [X5: num] :
      ( ( ord_less_num @ X5 @ Z )
     => ~ ( ord_less_eq_num @ T @ X5 ) ) ).

% minf(8)
thf(fact_2343_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X5: nat] :
      ( ( ord_less_nat @ X5 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X5 ) ) ).

% minf(8)
thf(fact_2344_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z: int] :
    ! [X5: int] :
      ( ( ord_less_int @ X5 @ Z )
     => ~ ( ord_less_eq_int @ T @ X5 ) ) ).

% minf(8)
thf(fact_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_complete__interval,axiom,
    ! [A: real,B: real,P2: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: real] :
              ( ( ord_less_eq_real @ A @ C )
              & ( ord_less_eq_real @ C @ B )
              & ! [X5: real] :
                  ( ( ( ord_less_eq_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D4: real] :
                  ( ! [X3: real] :
                      ( ( ( ord_less_eq_real @ A @ X3 )
                        & ( ord_less_real @ X3 @ D4 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_real @ D4 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_2356_complete__interval,axiom,
    ! [A: nat,B: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B )
              & ! [X5: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X5 )
                    & ( ord_less_nat @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D4: nat] :
                  ( ! [X3: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X3 )
                        & ( ord_less_nat @ X3 @ D4 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_nat @ D4 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_2357_complete__interval,axiom,
    ! [A: int,B: int,P2: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B )
         => ? [C: int] :
              ( ( ord_less_eq_int @ A @ C )
              & ( ord_less_eq_int @ C @ B )
              & ! [X5: int] :
                  ( ( ( ord_less_eq_int @ A @ X5 )
                    & ( ord_less_int @ X5 @ C ) )
                 => ( P2 @ X5 ) )
              & ! [D4: int] :
                  ( ! [X3: int] :
                      ( ( ( ord_less_eq_int @ A @ X3 )
                        & ( ord_less_int @ X3 @ D4 ) )
                     => ( P2 @ X3 ) )
                 => ( ord_less_eq_int @ D4 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_2358_vebt__insert_Osimps_I4_J,axiom,
    ! [V2: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary ) @ X )
      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ X ) ) @ ( suc @ ( suc @ V2 ) ) @ TreeList2 @ Summary ) ) ).

% vebt_insert.simps(4)
thf(fact_2359_nth__zip,axiom,
    ! [I2: nat,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( ord_less_nat @ I2 @ ( size_s3445333598471063425nteger @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3445333598471063425nteger @ Ys ) )
       => ( ( nth_Pr2304437835452373666nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) @ I2 )
          = ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ I2 ) @ ( nth_Code_integer @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2360_nth__zip,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Ys ) @ I2 )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_VEBT_VEBT @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2361_nth__zip,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs2 @ Ys ) @ I2 )
          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_o @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2362_nth__zip,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs2 @ Ys ) @ I2 )
          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2363_nth__zip,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs2 @ Ys ) @ I2 )
          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) @ ( nth_int @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2364_nth__zip,axiom,
    ! [I2: nat,Xs2: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs2 @ Ys ) @ I2 )
          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ I2 ) @ ( nth_VEBT_VEBT @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2365_nth__zip,axiom,
    ! [I2: nat,Xs2: list_o,Ys: list_o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs2 @ Ys ) @ I2 )
          = ( product_Pair_o_o @ ( nth_o @ Xs2 @ I2 ) @ ( nth_o @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2366_nth__zip,axiom,
    ! [I2: nat,Xs2: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs2 @ Ys ) @ I2 )
          = ( product_Pair_o_nat @ ( nth_o @ Xs2 @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2367_nth__zip,axiom,
    ! [I2: nat,Xs2: list_o,Ys: list_int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs2 @ Ys ) @ I2 )
          = ( product_Pair_o_int @ ( nth_o @ Xs2 @ I2 ) @ ( nth_int @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2368_nth__zip,axiom,
    ! [I2: nat,Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs2 @ Ys ) @ I2 )
          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ I2 ) @ ( nth_VEBT_VEBT @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_2369_find__Some__iff2,axiom,
    ! [X: product_prod_nat_nat,P2: product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X )
        = ( find_P8199882355184865565at_nat @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
            & ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2370_find__Some__iff2,axiom,
    ! [X: num,P2: num > $o,Xs2: list_num] :
      ( ( ( some_num @ X )
        = ( find_num @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_num @ Xs2 ) )
            & ( P2 @ ( nth_num @ Xs2 @ I ) )
            & ( X
              = ( nth_num @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_num @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2371_find__Some__iff2,axiom,
    ! [X: vEBT_VEBT,P2: vEBT_VEBT > $o,Xs2: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X )
        = ( find_VEBT_VEBT @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2372_find__Some__iff2,axiom,
    ! [X: $o,P2: $o > $o,Xs2: list_o] :
      ( ( ( some_o @ X )
        = ( find_o @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
            & ( P2 @ ( nth_o @ Xs2 @ I ) )
            & ( X
              = ( nth_o @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_o @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2373_find__Some__iff2,axiom,
    ! [X: nat,P2: nat > $o,Xs2: list_nat] :
      ( ( ( some_nat @ X )
        = ( find_nat @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
            & ( P2 @ ( nth_nat @ Xs2 @ I ) )
            & ( X
              = ( nth_nat @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2374_find__Some__iff2,axiom,
    ! [X: int,P2: int > $o,Xs2: list_int] :
      ( ( ( some_int @ X )
        = ( find_int @ P2 @ Xs2 ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
            & ( P2 @ ( nth_int @ Xs2 @ I ) )
            & ( X
              = ( nth_int @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_int @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_2375_find__Some__iff,axiom,
    ! [P2: product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat,X: product_prod_nat_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P2 @ Xs2 )
        = ( some_P7363390416028606310at_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5460976970255530739at_nat @ Xs2 ) )
            & ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
            & ( X
              = ( nth_Pr7617993195940197384at_nat @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2376_find__Some__iff,axiom,
    ! [P2: num > $o,Xs2: list_num,X: num] :
      ( ( ( find_num @ P2 @ Xs2 )
        = ( some_num @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_num @ Xs2 ) )
            & ( P2 @ ( nth_num @ Xs2 @ I ) )
            & ( X
              = ( nth_num @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_num @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2377_find__Some__iff,axiom,
    ! [P2: vEBT_VEBT > $o,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P2 @ Xs2 )
        = ( some_VEBT_VEBT @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ I ) )
            & ( X
              = ( nth_VEBT_VEBT @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_VEBT_VEBT @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2378_find__Some__iff,axiom,
    ! [P2: $o > $o,Xs2: list_o,X: $o] :
      ( ( ( find_o @ P2 @ Xs2 )
        = ( some_o @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs2 ) )
            & ( P2 @ ( nth_o @ Xs2 @ I ) )
            & ( X
              = ( nth_o @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_o @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2379_find__Some__iff,axiom,
    ! [P2: nat > $o,Xs2: list_nat,X: nat] :
      ( ( ( find_nat @ P2 @ Xs2 )
        = ( some_nat @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
            & ( P2 @ ( nth_nat @ Xs2 @ I ) )
            & ( X
              = ( nth_nat @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2380_find__Some__iff,axiom,
    ! [P2: int > $o,Xs2: list_int,X: int] :
      ( ( ( find_int @ P2 @ Xs2 )
        = ( some_int @ X ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs2 ) )
            & ( P2 @ ( nth_int @ Xs2 @ I ) )
            & ( X
              = ( nth_int @ Xs2 @ I ) )
            & ! [J: nat] :
                ( ( ord_less_nat @ J @ I )
               => ~ ( P2 @ ( nth_int @ Xs2 @ J ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_2381_nth__Cons__pos,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ N )
        = ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_2382_nth__Cons__pos,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
        = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_2383_rotate1__length01,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ one_one_nat )
     => ( ( rotate1_VEBT_VEBT @ Xs2 )
        = Xs2 ) ) ).

% rotate1_length01
thf(fact_2384_rotate1__length01,axiom,
    ! [Xs2: list_o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ one_one_nat )
     => ( ( rotate1_o @ Xs2 )
        = Xs2 ) ) ).

% rotate1_length01
thf(fact_2385_rotate1__length01,axiom,
    ! [Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ one_one_nat )
     => ( ( rotate1_nat @ Xs2 )
        = Xs2 ) ) ).

% rotate1_length01
thf(fact_2386_rotate1__length01,axiom,
    ! [Xs2: list_int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ one_one_nat )
     => ( ( rotate1_int @ Xs2 )
        = Xs2 ) ) ).

% rotate1_length01
thf(fact_2387_length__rotate1,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( rotate1_VEBT_VEBT @ Xs2 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_rotate1
thf(fact_2388_length__rotate1,axiom,
    ! [Xs2: list_o] :
      ( ( size_size_list_o @ ( rotate1_o @ Xs2 ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_rotate1
thf(fact_2389_length__rotate1,axiom,
    ! [Xs2: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_rotate1
thf(fact_2390_length__rotate1,axiom,
    ! [Xs2: list_int] :
      ( ( size_size_list_int @ ( rotate1_int @ Xs2 ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_rotate1
thf(fact_2391_nth__Cons__Suc,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ ( suc @ N ) )
      = ( nth_VEBT_VEBT @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_2392_nth__Cons__Suc,axiom,
    ! [X: nat,Xs2: list_nat,N: nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ N ) )
      = ( nth_nat @ Xs2 @ N ) ) ).

% nth_Cons_Suc
thf(fact_2393_nth__Cons__0,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_2394_nth__Cons__0,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_2395_zip__Cons__Cons,axiom,
    ! [X: code_integer,Xs2: list_Code_integer,Y: code_integer,Ys: list_Code_integer] :
      ( ( zip_Co3543743374963494515nteger @ ( cons_Code_integer @ X @ Xs2 ) @ ( cons_Code_integer @ Y @ Ys ) )
      = ( cons_P9044669534377732177nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_2396_zip__Cons__Cons,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) )
      = ( cons_P8732206157123786781at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_2397_zip__Cons__Cons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( zip_se5600341670672612855at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) )
      = ( cons_P3940603068885512221at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( zip_se5600341670672612855at_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_2398_zip__Cons__Cons,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_2399_zip__Cons__Cons,axiom,
    ! [X: int,Xs2: list_int,Y: int,Ys: list_int] :
      ( ( zip_int_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) )
      = ( cons_P3334398858971670639nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( zip_int_int @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_2400_enumerate__simps_I2_J,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( enumerate_nat @ N @ ( cons_nat @ X @ Xs2 ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ N @ X ) @ ( enumerate_nat @ ( suc @ N ) @ Xs2 ) ) ) ).

% enumerate_simps(2)
thf(fact_2401_zip__eq__ConsE,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,Xy: produc8923325533196201883nteger,Xys: list_P5578671422887162913nteger] :
      ( ( ( zip_Co3543743374963494515nteger @ Xs2 @ Ys )
        = ( cons_P9044669534377732177nteger @ Xy @ Xys ) )
     => ~ ! [X3: code_integer,Xs4: list_Code_integer] :
            ( ( Xs2
              = ( cons_Code_integer @ X3 @ Xs4 ) )
           => ! [Y3: code_integer,Ys4: list_Code_integer] :
                ( ( Ys
                  = ( cons_Code_integer @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc1086072967326762835nteger @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Co3543743374963494515nteger @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_2402_zip__eq__ConsE,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xy: produc859450856879609959at_nat,Xys: list_P8469869581646625389at_nat] :
      ( ( ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys )
        = ( cons_P8732206157123786781at_nat @ Xy @ Xys ) )
     => ~ ! [X3: product_prod_nat_nat,Xs4: list_P6011104703257516679at_nat] :
            ( ( Xs2
              = ( cons_P6512896166579812791at_nat @ X3 @ Xs4 ) )
           => ! [Y3: product_prod_nat_nat,Ys4: list_P6011104703257516679at_nat] :
                ( ( Ys
                  = ( cons_P6512896166579812791at_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6161850002892822231at_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_Pr4664179122662387191at_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_2403_zip__eq__ConsE,axiom,
    ! [Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat,Xy: produc3843707927480180839at_nat,Xys: list_P5464809261938338413at_nat] :
      ( ( ( zip_se5600341670672612855at_nat @ Xs2 @ Ys )
        = ( cons_P3940603068885512221at_nat @ Xy @ Xys ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Xs4: list_s1210847774152347623at_nat] :
            ( ( Xs2
              = ( cons_s6881495754146722583at_nat @ X3 @ Xs4 ) )
           => ! [Y3: set_Pr1261947904930325089at_nat,Ys4: list_s1210847774152347623at_nat] :
                ( ( Ys
                  = ( cons_s6881495754146722583at_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc2922128104949294807at_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_se5600341670672612855at_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_2404_zip__eq__ConsE,axiom,
    ! [Xs2: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs2 @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat,Xs4: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_2405_zip__eq__ConsE,axiom,
    ! [Xs2: list_int,Ys: list_int,Xy: product_prod_int_int,Xys: list_P5707943133018811711nt_int] :
      ( ( ( zip_int_int @ Xs2 @ Ys )
        = ( cons_P3334398858971670639nt_int @ Xy @ Xys ) )
     => ~ ! [X3: int,Xs4: list_int] :
            ( ( Xs2
              = ( cons_int @ X3 @ Xs4 ) )
           => ! [Y3: int,Ys4: list_int] :
                ( ( Ys
                  = ( cons_int @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_int_int @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_int_int @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_2406_find_Osimps_I2_J,axiom,
    ! [P2: product_prod_nat_nat > $o,X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( ( P2 @ X )
       => ( ( find_P8199882355184865565at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
          = ( some_P7363390416028606310at_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_P8199882355184865565at_nat @ P2 @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) )
          = ( find_P8199882355184865565at_nat @ P2 @ Xs2 ) ) ) ) ).

% find.simps(2)
thf(fact_2407_find_Osimps_I2_J,axiom,
    ! [P2: nat > $o,X: nat,Xs2: list_nat] :
      ( ( ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs2 ) )
          = ( some_nat @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_nat @ P2 @ ( cons_nat @ X @ Xs2 ) )
          = ( find_nat @ P2 @ Xs2 ) ) ) ) ).

% find.simps(2)
thf(fact_2408_find_Osimps_I2_J,axiom,
    ! [P2: num > $o,X: num,Xs2: list_num] :
      ( ( ( P2 @ X )
       => ( ( find_num @ P2 @ ( cons_num @ X @ Xs2 ) )
          = ( some_num @ X ) ) )
      & ( ~ ( P2 @ X )
       => ( ( find_num @ P2 @ ( cons_num @ X @ Xs2 ) )
          = ( find_num @ P2 @ Xs2 ) ) ) ) ).

% find.simps(2)
thf(fact_2409_set__zip__rightD,axiom,
    ! [X: code_integer,Y: code_integer,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) )
     => ( member_Code_integer @ Y @ ( set_Code_integer2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_2410_set__zip__rightD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys ) ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_2411_set__zip__rightD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs2 @ Ys ) ) )
     => ( member2643936169264416010at_nat @ Y @ ( set_se5049602875457034614at_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_2412_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_2413_set__zip__rightD,axiom,
    ! [X: int,Y: int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
     => ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_2414_set__zip__leftD,axiom,
    ! [X: code_integer,Y: code_integer,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) )
     => ( member_Code_integer @ X @ ( set_Code_integer2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_2415_set__zip__leftD,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys ) ) )
     => ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_2416_set__zip__leftD,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs2 @ Ys ) ) )
     => ( member2643936169264416010at_nat @ X @ ( set_se5049602875457034614at_nat @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_2417_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_2418_set__zip__leftD,axiom,
    ! [X: int,Y: int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Ys ) ) )
     => ( member_int @ X @ ( set_int2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_2419_in__set__zipE,axiom,
    ! [X: complex,Y: complex,Xs2: list_complex,Ys: list_complex] :
      ( ( member5793383173714906214omplex @ ( produc101793102246108661omplex @ X @ Y ) @ ( set_Pr8199049879907524818omplex @ ( zip_complex_complex @ Xs2 @ Ys ) ) )
     => ~ ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
         => ~ ( member_complex @ Y @ ( set_complex2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2420_in__set__zipE,axiom,
    ! [X: complex,Y: real,Xs2: list_complex,Ys: list_real] :
      ( ( member47443559803733732x_real @ ( produc1746590499379883635x_real @ X @ Y ) @ ( set_Pr1225976482156248400x_real @ ( zip_complex_real @ Xs2 @ Ys ) ) )
     => ~ ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
         => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2421_in__set__zipE,axiom,
    ! [X: complex,Y: int,Xs2: list_complex,Ys: list_int] :
      ( ( member595073364599660772ex_int @ ( produc1367138851071493491ex_int @ X @ Y ) @ ( set_Pr4995810437751016784ex_int @ ( zip_complex_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2422_in__set__zipE,axiom,
    ! [X: real,Y: complex,Xs2: list_real,Ys: list_complex] :
      ( ( member7358116576843751780omplex @ ( produc1693001998875562995omplex @ X @ Y ) @ ( set_Pr8536649499196266448omplex @ ( zip_real_complex @ Xs2 @ Ys ) ) )
     => ~ ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
         => ~ ( member_complex @ Y @ ( set_complex2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2423_in__set__zipE,axiom,
    ! [X: real,Y: real,Xs2: list_real,Ys: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X @ Y ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs2 @ Ys ) ) )
     => ~ ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
         => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2424_in__set__zipE,axiom,
    ! [X: real,Y: int,Xs2: list_real,Ys: list_int] :
      ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs2 @ Ys ) ) )
     => ~ ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
         => ~ ( member_int @ Y @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2425_in__set__zipE,axiom,
    ! [X: int,Y: complex,Xs2: list_int,Ys: list_complex] :
      ( ( member8811922270175639012omplex @ ( produc7948753499206759283omplex @ X @ Y ) @ ( set_Pr3989287306472219216omplex @ ( zip_int_complex @ Xs2 @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
         => ~ ( member_complex @ Y @ ( set_complex2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2426_in__set__zipE,axiom,
    ! [X: int,Y: real,Xs2: list_int,Ys: list_real] :
      ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X @ Y ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs2 @ Ys ) ) )
     => ~ ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
         => ~ ( member_real @ Y @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2427_in__set__zipE,axiom,
    ! [X: complex,Y: vEBT_VEBT,Xs2: list_complex,Ys: list_VEBT_VEBT] :
      ( ( member1978952105866562066T_VEBT @ ( produc2757191886755552429T_VEBT @ X @ Y ) @ ( set_Pr5158653123227461798T_VEBT @ ( zip_co9157518722488180109T_VEBT @ Xs2 @ Ys ) ) )
     => ~ ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
         => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2428_in__set__zipE,axiom,
    ! [X: real,Y: vEBT_VEBT,Xs2: list_real,Ys: list_VEBT_VEBT] :
      ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs2 @ Ys ) ) )
     => ~ ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
         => ~ ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_2429_zip__same,axiom,
    ! [A: complex,B: complex,Xs2: list_complex] :
      ( ( member5793383173714906214omplex @ ( produc101793102246108661omplex @ A @ B ) @ ( set_Pr8199049879907524818omplex @ ( zip_complex_complex @ Xs2 @ Xs2 ) ) )
      = ( ( member_complex @ A @ ( set_complex2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2430_zip__same,axiom,
    ! [A: real,B: real,Xs2: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs2 @ Xs2 ) ) )
      = ( ( member_real @ A @ ( set_real2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2431_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs2: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2432_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Xs2 ) ) )
      = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2433_zip__same,axiom,
    ! [A: code_integer,B: code_integer,Xs2: list_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ A @ B ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Xs2 ) ) )
      = ( ( member_Code_integer @ A @ ( set_Code_integer2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2434_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2435_zip__same,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ A @ B ) @ ( set_Pr3765526544606949372at_nat @ ( zip_se5600341670672612855at_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member2643936169264416010at_nat @ A @ ( set_se5049602875457034614at_nat @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2436_zip__same,axiom,
    ! [A: nat,B: nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Xs2 ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2437_zip__same,axiom,
    ! [A: int,B: int,Xs2: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs2 @ Xs2 ) ) )
      = ( ( member_int @ A @ ( set_int2 @ Xs2 ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_2438_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ( suc @ N )
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
      = ( ? [Y4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs2
              = ( cons_VEBT_VEBT @ Y4 @ Ys3 ) )
            & ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_2439_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ( suc @ N )
        = ( size_size_list_o @ Xs2 ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_2440_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ( suc @ N )
        = ( size_size_list_nat @ Xs2 ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_2441_Suc__length__conv,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ( suc @ N )
        = ( size_size_list_int @ Xs2 ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% Suc_length_conv
thf(fact_2442_length__Suc__conv,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs2
              = ( cons_VEBT_VEBT @ Y4 @ Ys3 ) )
            & ( ( size_s6755466524823107622T_VEBT @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_2443_length__Suc__conv,axiom,
    ! [Xs2: list_o,N: nat] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ Y4 @ Ys3 ) )
            & ( ( size_size_list_o @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_2444_length__Suc__conv,axiom,
    ! [Xs2: list_nat,N: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys3 ) )
            & ( ( size_size_list_nat @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_2445_length__Suc__conv,axiom,
    ! [Xs2: list_int,N: nat] :
      ( ( ( size_size_list_int @ Xs2 )
        = ( suc @ N ) )
      = ( ? [Y4: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ Y4 @ Ys3 ) )
            & ( ( size_size_list_int @ Ys3 )
              = N ) ) ) ) ).

% length_Suc_conv
thf(fact_2446_impossible__Cons,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( Xs2
       != ( cons_VEBT_VEBT @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_2447_impossible__Cons,axiom,
    ! [Xs2: list_o,Ys: list_o,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ ( size_size_list_o @ Ys ) )
     => ( Xs2
       != ( cons_o @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_2448_impossible__Cons,axiom,
    ! [Xs2: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs2
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_2449_impossible__Cons,axiom,
    ! [Xs2: list_int,Ys: list_int,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ ( size_size_list_int @ Ys ) )
     => ( Xs2
       != ( cons_int @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_2450_vebt__insert_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info2 @ zero_zero_nat @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(2)
thf(fact_2451_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,X: code_integer] :
      ( ( ( size_s3445333598471063425nteger @ Xs2 )
        = ( size_s3445333598471063425nteger @ Ys ) )
     => ( ( member_Code_integer @ X @ ( set_Code_integer2 @ Xs2 ) )
       => ~ ! [Y3: code_integer] :
              ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y3 ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2452_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_complex,Ys: list_VEBT_VEBT,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member1978952105866562066T_VEBT @ ( produc2757191886755552429T_VEBT @ X @ Y3 ) @ ( set_Pr5158653123227461798T_VEBT @ ( zip_co9157518722488180109T_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2453_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_VEBT_VEBT,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2454_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_complex,Ys: list_o,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ~ ! [Y3: $o] :
              ~ ( member6487239523555734774plex_o @ ( produc2908979694703026321plex_o @ X @ Y3 ) @ ( set_Pr6829704231520703882plex_o @ ( zip_complex_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2455_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_o,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: $o] :
              ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X @ Y3 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2456_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_complex,Ys: list_nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member4772924384108857480ex_nat @ ( produc1369629321580543767ex_nat @ X @ Y3 ) @ ( set_Pr9173661457260213492ex_nat @ ( zip_complex_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2457_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: nat] :
              ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2458_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_complex,Ys: list_int,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ~ ! [Y3: int] :
              ~ ( member595073364599660772ex_int @ ( produc1367138851071493491ex_int @ X @ Y3 ) @ ( set_Pr4995810437751016784ex_int @ ( zip_complex_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2459_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_real,Ys: list_int,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ~ ! [Y3: int] :
              ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2460_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ~ ! [Y3: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2461_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,Y: code_integer] :
      ( ( ( size_s3445333598471063425nteger @ Xs2 )
        = ( size_s3445333598471063425nteger @ Ys ) )
     => ( ( member_Code_integer @ Y @ ( set_Code_integer2 @ Ys ) )
       => ~ ! [X3: code_integer] :
              ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y ) @ ( set_Pr920681315882439344nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2462_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_complex,Y: complex] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s3451745648224563538omplex @ Ys ) )
     => ( ( member_complex @ Y @ ( set_complex2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member3207599676835851048omplex @ ( produc5617778602380981643omplex @ X3 @ Y ) @ ( set_Pr6387300694196750780omplex @ ( zip_VE2794733401258833515omplex @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2463_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_real,Y: real] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X3 @ Y ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2464_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2465_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o,Y: $o] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X3 @ Y ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2466_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,Y: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2467_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int,Y: int] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y @ ( set_int2 @ Ys ) )
       => ~ ! [X3: vEBT_VEBT] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2468_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_complex,Y: complex] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_s3451745648224563538omplex @ Ys ) )
     => ( ( member_complex @ Y @ ( set_complex2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member1046615901120239500omplex @ ( produc414345526774272751omplex @ X3 @ Y ) @ ( set_Pr1389080609085208608omplex @ ( zip_o_complex @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2469_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_real,Y: real] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y @ ( set_real2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X3 @ Y ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2470_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( ( size_size_list_o @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X3: $o] :
              ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X3 @ Y ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2471_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
      = ( ? [X4: vEBT_VEBT,Ys3: list_VEBT_VEBT] :
            ( ( Xs2
              = ( cons_VEBT_VEBT @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_2472_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs2 ) )
      = ( ? [X4: $o,Ys3: list_o] :
            ( ( Xs2
              = ( cons_o @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_o @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_2473_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) )
      = ( ? [X4: nat,Ys3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_2474_Suc__le__length__iff,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs2 ) )
      = ( ? [X4: int,Ys3: list_int] :
            ( ( Xs2
              = ( cons_int @ X4 @ Ys3 ) )
            & ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys3 ) ) ) ) ) ).

% Suc_le_length_iff
thf(fact_2475_VEBT__internal_Oinsert_H_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X: nat] :
      ( ( vEBT_VEBT_insert @ ( vEBT_Leaf @ A @ B ) @ X )
      = ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X ) ) ).

% VEBT_internal.insert'.simps(1)
thf(fact_2476_vebt__insert_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Ts2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) @ X )
      = ( vEBT_Node @ Info2 @ ( suc @ zero_zero_nat ) @ Ts2 @ S2 ) ) ).

% vebt_insert.simps(3)
thf(fact_2477_vebt__insert_Osimps_I1_J,axiom,
    ! [X: nat,A: $o,B: $o] :
      ( ( ( X = zero_zero_nat )
       => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
          = ( vEBT_Leaf @ $true @ B ) ) )
      & ( ( X != zero_zero_nat )
       => ( ( ( X = one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ $true ) ) )
          & ( ( X != one_one_nat )
           => ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A @ B ) @ X )
              = ( vEBT_Leaf @ A @ B ) ) ) ) ) ) ).

% vebt_insert.simps(1)
thf(fact_2478_list_Osize_I4_J,axiom,
    ! [X21: vEBT_VEBT,X22: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_s6755466524823107622T_VEBT @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_2479_list_Osize_I4_J,axiom,
    ! [X21: $o,X22: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_o @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_2480_list_Osize_I4_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_2481_list_Osize_I4_J,axiom,
    ! [X21: int,X22: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
      = ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% list.size(4)
thf(fact_2482_nth__Cons_H,axiom,
    ! [N: nat,X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ N )
          = ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_2483_nth__Cons_H,axiom,
    ! [N: nat,X: nat,Xs2: list_nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = X ) )
      & ( ( N != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_2484_nth__equal__first__eq,axiom,
    ! [X: complex,Xs2: list_complex,N: nat] :
      ( ~ ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
       => ( ( ( nth_complex @ ( cons_complex @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2485_nth__equal__first__eq,axiom,
    ! [X: real,Xs2: list_real,N: nat] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( ( nth_real @ ( cons_real @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2486_nth__equal__first__eq,axiom,
    ! [X: set_nat,Xs2: list_set_nat,N: nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
       => ( ( ( nth_set_nat @ ( cons_set_nat @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2487_nth__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2488_nth__equal__first__eq,axiom,
    ! [X: $o,Xs2: list_o,N: nat] :
      ( ~ ( member_o @ X @ ( set_o2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( ( nth_o @ ( cons_o @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2489_nth__equal__first__eq,axiom,
    ! [X: nat,Xs2: list_nat,N: nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2490_nth__equal__first__eq,axiom,
    ! [X: int,Xs2: list_int,N: nat] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ( ord_less_eq_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( ( nth_int @ ( cons_int @ X @ Xs2 ) @ N )
            = X )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_2491_nth__non__equal__first__eq,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_VEBT_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_VEBT_VEBT @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_2492_nth__non__equal__first__eq,axiom,
    ! [X: nat,Y: nat,Xs2: list_nat,N: nat] :
      ( ( X != Y )
     => ( ( ( nth_nat @ ( cons_nat @ X @ Xs2 ) @ N )
          = Y )
        = ( ( ( nth_nat @ Xs2 @ ( minus_minus_nat @ N @ one_one_nat ) )
            = Y )
          & ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_2493_length__Cons,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) )
      = ( suc @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% length_Cons
thf(fact_2494_length__Cons,axiom,
    ! [X: $o,Xs2: list_o] :
      ( ( size_size_list_o @ ( cons_o @ X @ Xs2 ) )
      = ( suc @ ( size_size_list_o @ Xs2 ) ) ) ).

% length_Cons
thf(fact_2495_length__Cons,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( size_size_list_nat @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( size_size_list_nat @ Xs2 ) ) ) ).

% length_Cons
thf(fact_2496_length__Cons,axiom,
    ! [X: int,Xs2: list_int] :
      ( ( size_size_list_int @ ( cons_int @ X @ Xs2 ) )
      = ( suc @ ( size_size_list_int @ Xs2 ) ) ) ).

% length_Cons
thf(fact_2497_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% dbl_inc_simps(2)
thf(fact_2498_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_2499_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_2500_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_2501_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_2502_enumerate__Suc_H,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) )
      = ( infini8530281810654367211te_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ ( infini8530281810654367211te_nat @ S3 @ zero_zero_nat ) @ bot_bot_set_nat ) ) @ N ) ) ).

% enumerate_Suc'
thf(fact_2503_dbl__dec__def,axiom,
    ( neg_nu7757733837767384882nteger
    = ( ^ [X4: code_integer] : ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ X4 @ X4 ) @ one_one_Code_integer ) ) ) ).

% dbl_dec_def
thf(fact_2504_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X4: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_2505_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_2506_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_2507_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_2508_Gcd__fin__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( semiri4256215615220890538in_int @ A4 )
        = zero_zero_int )
      = ( ( ord_less_eq_set_int @ A4 @ ( insert_int @ zero_zero_int @ bot_bot_set_int ) )
        & ( finite_finite_int @ A4 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_2509_Gcd__fin__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( semiri4258706085729940814in_nat @ A4 )
        = zero_zero_nat )
      = ( ( ord_less_eq_set_nat @ A4 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) )
        & ( finite_finite_nat @ A4 ) ) ) ).

% Gcd_fin_0_iff
thf(fact_2510_power__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).

% power_decreasing_iff
thf(fact_2511_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_2512_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_2513_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_2514_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_2515_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_real @ ( insert_real @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2516_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_nat @ ( insert_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2517_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_complex,X: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_complex @ ( insert_complex @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2518_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_int,X: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_int @ ( insert_int @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2519_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_list_nat,X: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( insert_list_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2520_card__insert__le__m1,axiom,
    ! [N: nat,Y: set_set_nat,X: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ Y ) @ ( minus_minus_nat @ N @ one_one_nat ) )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( insert_set_nat @ X @ Y ) ) @ N ) ) ) ).

% card_insert_le_m1
thf(fact_2521_count__notin,axiom,
    ! [X: complex,Xs2: list_complex] :
      ( ~ ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
     => ( ( count_list_complex @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2522_count__notin,axiom,
    ! [X: real,Xs2: list_real] :
      ( ~ ( member_real @ X @ ( set_real2 @ Xs2 ) )
     => ( ( count_list_real @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2523_count__notin,axiom,
    ! [X: set_nat,Xs2: list_set_nat] :
      ( ~ ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
     => ( ( count_list_set_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2524_count__notin,axiom,
    ! [X: int,Xs2: list_int] :
      ( ~ ( member_int @ X @ ( set_int2 @ Xs2 ) )
     => ( ( count_list_int @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2525_count__notin,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
     => ( ( count_list_VEBT_VEBT @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2526_count__notin,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
     => ( ( count_list_nat @ Xs2 @ X )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_2527_Cons__lenlex__iff,axiom,
    ! [M2: code_integer,Ms: list_Code_integer,N: code_integer,Ns: list_Code_integer,R3: set_Pr4811707699266497531nteger] :
      ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ ( cons_Code_integer @ M2 @ Ms ) @ ( cons_Code_integer @ N @ Ns ) ) @ ( lenlex_Code_integer @ R3 ) )
      = ( ( ord_less_nat @ ( size_s3445333598471063425nteger @ Ms ) @ ( size_s3445333598471063425nteger @ Ns ) )
        | ( ( ( size_s3445333598471063425nteger @ Ms )
            = ( size_s3445333598471063425nteger @ Ns ) )
          & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Ms @ Ns ) @ ( lenlex_Code_integer @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2528_Cons__lenlex__iff,axiom,
    ! [M2: product_prod_nat_nat,Ms: list_P6011104703257516679at_nat,N: product_prod_nat_nat,Ns: list_P6011104703257516679at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ M2 @ Ms ) @ ( cons_P6512896166579812791at_nat @ N @ Ns ) ) @ ( lenlex325483962726685836at_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_s5460976970255530739at_nat @ Ms ) @ ( size_s5460976970255530739at_nat @ Ns ) )
        | ( ( ( size_s5460976970255530739at_nat @ Ms )
            = ( size_s5460976970255530739at_nat @ Ns ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ms @ Ns ) @ ( lenlex325483962726685836at_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2529_Cons__lenlex__iff,axiom,
    ! [M2: set_Pr1261947904930325089at_nat,Ms: list_s1210847774152347623at_nat,N: set_Pr1261947904930325089at_nat,Ns: list_s1210847774152347623at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ M2 @ Ms ) @ ( cons_s6881495754146722583at_nat @ N @ Ns ) ) @ ( lenlex1357538814655152620at_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_s8736152011456118867at_nat @ Ms ) @ ( size_s8736152011456118867at_nat @ Ns ) )
        | ( ( ( size_s8736152011456118867at_nat @ Ms )
            = ( size_s8736152011456118867at_nat @ Ns ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Ms @ Ns ) @ ( lenlex1357538814655152620at_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2530_Cons__lenlex__iff,axiom,
    ! [M2: vEBT_VEBT,Ms: list_VEBT_VEBT,N: vEBT_VEBT,Ns: list_VEBT_VEBT,R3: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ ( cons_VEBT_VEBT @ M2 @ Ms ) @ ( cons_VEBT_VEBT @ N @ Ns ) ) @ ( lenlex_VEBT_VEBT @ R3 ) )
      = ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) )
        | ( ( ( size_s6755466524823107622T_VEBT @ Ms )
            = ( size_s6755466524823107622T_VEBT @ Ns ) )
          & ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2531_Cons__lenlex__iff,axiom,
    ! [M2: $o,Ms: list_o,N: $o,Ns: list_o,R3: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ M2 @ Ms ) @ ( cons_o @ N @ Ns ) ) @ ( lenlex_o @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) )
        | ( ( ( size_size_list_o @ Ms )
            = ( size_size_list_o @ Ns ) )
          & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2532_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2533_Cons__lenlex__iff,axiom,
    ! [M2: int,Ms: list_int,N: int,Ns: list_int,R3: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
        | ( ( ( size_size_list_int @ Ms )
            = ( size_size_list_int @ Ns ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_2534_power__shift,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ( power_power_nat @ X @ Y )
        = Z3 )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X ) @ ( some_nat @ Y ) )
        = ( some_nat @ Z3 ) ) ) ).

% power_shift
thf(fact_2535_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_2536_nat__power__eq__Suc__0__iff,axiom,
    ! [X: nat,M2: nat] :
      ( ( ( power_power_nat @ X @ M2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2 = zero_zero_nat )
        | ( X
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2537_power__Suc__0,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2538_nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_2539_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% dbl_dec_simps(3)
thf(fact_2540_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_2541_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_2542_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_2543_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2544_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2545_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2546_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2547_power__0__Suc,axiom,
    ! [N: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2548_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2549_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2550_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2551_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2552_card_Oempty,axiom,
    ( ( finite_card_complex @ bot_bot_set_complex )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2553_card_Oempty,axiom,
    ( ( finite_card_list_nat @ bot_bot_set_list_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2554_card_Oempty,axiom,
    ( ( finite_card_set_nat @ bot_bot_set_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2555_card_Oempty,axiom,
    ( ( finite_card_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2556_card_Oempty,axiom,
    ( ( finite_card_int @ bot_bot_set_int )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2557_card_Oempty,axiom,
    ( ( finite_card_real @ bot_bot_set_real )
    = zero_zero_nat ) ).

% card.empty
thf(fact_2558_card_Oinfinite,axiom,
    ! [A4: set_list_nat] :
      ( ~ ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite_card_list_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_2559_card_Oinfinite,axiom,
    ! [A4: set_set_nat] :
      ( ~ ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite_card_set_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_2560_card_Oinfinite,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( finite_card_nat @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_2561_card_Oinfinite,axiom,
    ! [A4: set_int] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( finite_card_int @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_2562_card_Oinfinite,axiom,
    ! [A4: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_card_complex @ A4 )
        = zero_zero_nat ) ) ).

% card.infinite
thf(fact_2563_Gcd__fin_Oempty,axiom,
    ( ( semiri4258706085729940814in_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Gcd_fin.empty
thf(fact_2564_Gcd__fin_Oempty,axiom,
    ( ( semiri4256215615220890538in_int @ bot_bot_set_int )
    = zero_zero_int ) ).

% Gcd_fin.empty
thf(fact_2565_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_2566_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_2567_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_2568_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_2569_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_2570_card__0__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ( finite_card_list_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_list_nat ) ) ) ).

% card_0_eq
thf(fact_2571_card__0__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ( finite_card_set_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_set_nat ) ) ) ).

% card_0_eq
thf(fact_2572_card__0__eq,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( finite_card_complex @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_complex ) ) ) ).

% card_0_eq
thf(fact_2573_card__0__eq,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( finite_card_nat @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_nat ) ) ) ).

% card_0_eq
thf(fact_2574_card__0__eq,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( finite_card_int @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_int ) ) ) ).

% card_0_eq
thf(fact_2575_card__0__eq,axiom,
    ! [A4: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( finite_card_real @ A4 )
          = zero_zero_nat )
        = ( A4 = bot_bot_set_real ) ) ) ).

% card_0_eq
thf(fact_2576_card__insert__disjoint,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
          = ( suc @ ( finite_card_real @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2577_card__insert__disjoint,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ~ ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_list_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2578_card__insert__disjoint,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ~ ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_set_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2579_card__insert__disjoint,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
          = ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2580_card__insert__disjoint,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
          = ( suc @ ( finite_card_int @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2581_card__insert__disjoint,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
          = ( suc @ ( finite_card_complex @ A4 ) ) ) ) ) ).

% card_insert_disjoint
thf(fact_2582_power__strict__decreasing__iff,axiom,
    ! [B: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ B @ one_one_Code_integer )
       => ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ B @ M2 ) @ ( power_8256067586552552935nteger @ B @ N ) )
          = ( ord_less_nat @ N @ M2 ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2583_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_2584_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_2585_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_2586_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_2587_power__increasing__iff,axiom,
    ! [B: code_integer,X: nat,Y: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ B )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ B @ X ) @ ( power_8256067586552552935nteger @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2588_power__increasing__iff,axiom,
    ! [B: real,X: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2589_power__increasing__iff,axiom,
    ! [B: rat,X: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2590_power__increasing__iff,axiom,
    ! [B: nat,X: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2591_power__increasing__iff,axiom,
    ! [B: int,X: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power_increasing_iff
thf(fact_2592_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_2593_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_2594_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_2595_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_2596_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_2597_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_2598_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_2599_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_2600_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_2601_power__commuting__commutes,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = ( times_times_complex @ Y @ X ) )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2602_power__commuting__commutes,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = ( times_times_real @ Y @ X ) )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2603_power__commuting__commutes,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = ( times_times_rat @ Y @ X ) )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ Y )
        = ( times_times_rat @ Y @ ( power_power_rat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2604_power__commuting__commutes,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = ( times_times_nat @ Y @ X ) )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2605_power__commuting__commutes,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = ( times_times_int @ Y @ X ) )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X @ N ) ) ) ) ).

% power_commuting_commutes
thf(fact_2606_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_2607_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_2608_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_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_lenlex__irreflexive,axiom,
    ! [R3: set_Pr4811707699266497531nteger,Xs2: list_Code_integer] :
      ( ! [X3: code_integer] :
          ~ ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ X3 ) @ R3 )
     => ~ ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs2 @ Xs2 ) @ ( lenlex_Code_integer @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_2622_lenlex__irreflexive,axiom,
    ! [R3: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat] :
      ( ! [X3: product_prod_nat_nat] :
          ~ ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ X3 ) @ R3 )
     => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Xs2 ) @ ( lenlex325483962726685836at_nat @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_2623_lenlex__irreflexive,axiom,
    ! [R3: set_Pr4329608150637261639at_nat,Xs2: list_s1210847774152347623at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat] :
          ~ ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ X3 ) @ R3 )
     => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs2 @ Xs2 ) @ ( lenlex1357538814655152620at_nat @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_2624_lenlex__irreflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R3 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Xs2 ) @ ( lenlex_nat @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_2625_lenlex__irreflexive,axiom,
    ! [R3: set_Pr958786334691620121nt_int,Xs2: list_int] :
      ( ! [X3: int] :
          ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R3 )
     => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Xs2 ) @ ( lenlex_int @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_finite__enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ord_less_nat @ ( suc @ N ) @ ( finite_card_nat @ S3 ) )
       => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) ) ) ) ) ).

% finite_enumerate_step
thf(fact_2635_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_real,R3: real > real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B6: real] :
                ( ( member_real @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: real,A22: real,B3: real] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_real @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_real @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2636_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_nat,R3: nat > real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B6: real] :
                ( ( member_real @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: nat,A22: nat,B3: real] :
              ( ( member_nat @ A1 @ A4 )
             => ( ( member_nat @ A22 @ A4 )
               => ( ( member_real @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_real @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2637_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_complex,R3: complex > real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ? [B6: real] :
                ( ( member_real @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: complex,A22: complex,B3: real] :
              ( ( member_complex @ A1 @ A4 )
             => ( ( member_complex @ A22 @ A4 )
               => ( ( member_real @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_real @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2638_card__le__if__inj__on__rel,axiom,
    ! [B5: set_real,A4: set_int,R3: int > real > $o] :
      ( ( finite_finite_real @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B6: real] :
                ( ( member_real @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: int,A22: int,B3: real] :
              ( ( member_int @ A1 @ A4 )
             => ( ( member_int @ A22 @ A4 )
               => ( ( member_real @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_real @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2639_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_real,R3: real > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B6: nat] :
                ( ( member_nat @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: real,A22: real,B3: nat] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2640_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_nat,R3: nat > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B6: nat] :
                ( ( member_nat @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: nat,A22: nat,B3: nat] :
              ( ( member_nat @ A1 @ A4 )
             => ( ( member_nat @ A22 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2641_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_complex,R3: complex > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ? [B6: nat] :
                ( ( member_nat @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: complex,A22: complex,B3: nat] :
              ( ( member_complex @ A1 @ A4 )
             => ( ( member_complex @ A22 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2642_card__le__if__inj__on__rel,axiom,
    ! [B5: set_nat,A4: set_int,R3: int > nat > $o] :
      ( ( finite_finite_nat @ B5 )
     => ( ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ? [B6: nat] :
                ( ( member_nat @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: int,A22: int,B3: nat] :
              ( ( member_int @ A1 @ A4 )
             => ( ( member_int @ A22 @ A4 )
               => ( ( member_nat @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2643_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_real,R3: real > int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ? [B6: int] :
                ( ( member_int @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: real,A22: real,B3: int] :
              ( ( member_real @ A1 @ A4 )
             => ( ( member_real @ A22 @ A4 )
               => ( ( member_int @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2644_card__le__if__inj__on__rel,axiom,
    ! [B5: set_int,A4: set_nat,R3: nat > int > $o] :
      ( ( finite_finite_int @ B5 )
     => ( ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ? [B6: int] :
                ( ( member_int @ B6 @ B5 )
                & ( R3 @ A3 @ B6 ) ) )
       => ( ! [A1: nat,A22: nat,B3: int] :
              ( ( member_nat @ A1 @ A4 )
             => ( ( member_nat @ A22 @ A4 )
               => ( ( member_int @ B3 @ B5 )
                 => ( ( R3 @ A1 @ B3 )
                   => ( ( R3 @ A22 @ B3 )
                     => ( A1 = A22 ) ) ) ) ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ) ).

% card_le_if_inj_on_rel
thf(fact_2645_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_2646_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_2647_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_2648_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_2649_card__insert__le,axiom,
    ! [A4: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ ( finite_card_real @ ( insert_real @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2650_card__insert__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ ( insert_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2651_card__insert__le,axiom,
    ! [A4: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ ( insert_complex @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2652_card__insert__le,axiom,
    ! [A4: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ ( insert_int @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2653_card__insert__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2654_card__insert__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) ) ) ).

% card_insert_le
thf(fact_2655_one__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ).

% one_le_power
thf(fact_2656_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_2657_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_2658_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_2659_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_2660_left__right__inverse__power,axiom,
    ! [X: code_integer,Y: code_integer,N: nat] :
      ( ( ( times_3573771949741848930nteger @ X @ Y )
        = one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ X @ N ) @ ( power_8256067586552552935nteger @ Y @ N ) )
        = one_one_Code_integer ) ) ).

% left_right_inverse_power
thf(fact_2661_left__right__inverse__power,axiom,
    ! [X: complex,Y: complex,N: nat] :
      ( ( ( times_times_complex @ X @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_2662_left__right__inverse__power,axiom,
    ! [X: real,Y: real,N: nat] :
      ( ( ( times_times_real @ X @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_2663_left__right__inverse__power,axiom,
    ! [X: rat,Y: rat,N: nat] :
      ( ( ( times_times_rat @ X @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_2664_left__right__inverse__power,axiom,
    ! [X: nat,Y: nat,N: nat] :
      ( ( ( times_times_nat @ X @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_2665_left__right__inverse__power,axiom,
    ! [X: int,Y: int,N: nat] :
      ( ( ( times_times_int @ X @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_2666_finite__enum__subset,axiom,
    ! [X8: set_nat,Y7: set_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( finite_card_nat @ X8 ) )
         => ( ( infini8530281810654367211te_nat @ X8 @ I3 )
            = ( infini8530281810654367211te_nat @ Y7 @ I3 ) ) )
     => ( ( finite_finite_nat @ X8 )
       => ( ( finite_finite_nat @ Y7 )
         => ( ( ord_less_eq_nat @ ( finite_card_nat @ X8 ) @ ( finite_card_nat @ Y7 ) )
           => ( ord_less_eq_set_nat @ X8 @ Y7 ) ) ) ) ) ).

% finite_enum_subset
thf(fact_2667_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_2668_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_2669_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_2670_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_2671_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_2672_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_2673_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_2674_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_2675_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_2676_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_2677_power__0,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% power_0
thf(fact_2678_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2679_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2680_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2681_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2682_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_2683_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_2684_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_2685_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_2686_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_2687_nat__power__less__imp__less,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% nat_power_less_imp_less
thf(fact_2688_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_2689_card__eq__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_list_nat )
        | ~ ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2690_card__eq__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_set_nat )
        | ~ ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2691_card__eq__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( ( finite_card_complex @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_complex )
        | ~ ( finite3207457112153483333omplex @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2692_card__eq__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_nat )
        | ~ ( finite_finite_nat @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2693_card__eq__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ( finite_card_int @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_int )
        | ~ ( finite_finite_int @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2694_card__eq__0__iff,axiom,
    ! [A4: set_real] :
      ( ( ( finite_card_real @ A4 )
        = zero_zero_nat )
      = ( ( A4 = bot_bot_set_real )
        | ~ ( finite_finite_real @ A4 ) ) ) ).

% card_eq_0_iff
thf(fact_2695_card__ge__0__finite,axiom,
    ! [A4: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
     => ( finite8100373058378681591st_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_2696_card__ge__0__finite,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A4 ) )
     => ( finite1152437895449049373et_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_2697_card__ge__0__finite,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
     => ( finite_finite_nat @ A4 ) ) ).

% card_ge_0_finite
thf(fact_2698_card__ge__0__finite,axiom,
    ! [A4: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
     => ( finite_finite_int @ A4 ) ) ).

% card_ge_0_finite
thf(fact_2699_card__ge__0__finite,axiom,
    ! [A4: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
     => ( finite3207457112153483333omplex @ A4 ) ) ).

% card_ge_0_finite
thf(fact_2700_card__insert__if,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
            = ( finite_card_real @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
            = ( suc @ ( finite_card_real @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2701_card__insert__if,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ( member_list_nat @ X @ A4 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
            = ( finite_card_list_nat @ A4 ) ) )
        & ( ~ ( member_list_nat @ X @ A4 )
         => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_list_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2702_card__insert__if,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ( member_set_nat @ X @ A4 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
            = ( finite_card_set_nat @ A4 ) ) )
        & ( ~ ( member_set_nat @ X @ A4 )
         => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_set_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2703_card__insert__if,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
            = ( finite_card_nat @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
            = ( suc @ ( finite_card_nat @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2704_card__insert__if,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
            = ( finite_card_int @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
            = ( suc @ ( finite_card_int @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2705_card__insert__if,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X @ A4 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
            = ( finite_card_complex @ A4 ) ) )
        & ( ~ ( member_complex @ X @ A4 )
         => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
            = ( suc @ ( finite_card_complex @ A4 ) ) ) ) ) ) ).

% card_insert_if
thf(fact_2706_card__Suc__eq__finite,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: real,B7: set_real] :
            ( ( A4
              = ( insert_real @ B4 @ B7 ) )
            & ~ ( member_real @ B4 @ B7 )
            & ( ( finite_card_real @ B7 )
              = K2 )
            & ( finite_finite_real @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2707_card__Suc__eq__finite,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: list_nat,B7: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ B4 @ B7 ) )
            & ~ ( member_list_nat @ B4 @ B7 )
            & ( ( finite_card_list_nat @ B7 )
              = K2 )
            & ( finite8100373058378681591st_nat @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2708_card__Suc__eq__finite,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: set_nat,B7: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ B4 @ B7 ) )
            & ~ ( member_set_nat @ B4 @ B7 )
            & ( ( finite_card_set_nat @ B7 )
              = K2 )
            & ( finite1152437895449049373et_nat @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2709_card__Suc__eq__finite,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: nat,B7: set_nat] :
            ( ( A4
              = ( insert_nat @ B4 @ B7 ) )
            & ~ ( member_nat @ B4 @ B7 )
            & ( ( finite_card_nat @ B7 )
              = K2 )
            & ( finite_finite_nat @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2710_card__Suc__eq__finite,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: int,B7: set_int] :
            ( ( A4
              = ( insert_int @ B4 @ B7 ) )
            & ~ ( member_int @ B4 @ B7 )
            & ( ( finite_card_int @ B7 )
              = K2 )
            & ( finite_finite_int @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2711_card__Suc__eq__finite,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: complex,B7: set_complex] :
            ( ( A4
              = ( insert_complex @ B4 @ B7 ) )
            & ~ ( member_complex @ B4 @ B7 )
            & ( ( finite_card_complex @ B7 )
              = K2 )
            & ( finite3207457112153483333omplex @ B7 ) ) ) ) ).

% card_Suc_eq_finite
thf(fact_2712_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_2713_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_2714_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_2715_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_2716_card__mono,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_2717_card__mono,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_2718_card__mono,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ) ).

% card_mono
thf(fact_2719_card__mono,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B5 ) ) ) ) ).

% card_mono
thf(fact_2720_card__mono,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ) ).

% card_mono
thf(fact_2721_card__seteq,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ( ord_le6045566169113846134st_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ B5 ) @ ( finite_card_list_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_2722_card__seteq,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ( ord_le6893508408891458716et_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ B5 ) @ ( finite_card_set_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_2723_card__seteq,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ B5 ) @ ( finite_card_int @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_2724_card__seteq,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ B5 ) @ ( finite_card_complex @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_2725_card__seteq,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ( ord_less_eq_set_nat @ A4 @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ B5 ) @ ( finite_card_nat @ A4 ) )
         => ( A4 = B5 ) ) ) ) ).

% card_seteq
thf(fact_2726_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_list_nat,C4: nat] :
      ( ! [G: set_list_nat] :
          ( ( ord_le6045566169113846134st_nat @ G @ F3 )
         => ( ( finite8100373058378681591st_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_list_nat @ G ) @ C4 ) ) )
     => ( ( finite8100373058378681591st_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_list_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_2727_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_set_nat,C4: nat] :
      ( ! [G: set_set_nat] :
          ( ( ord_le6893508408891458716et_nat @ G @ F3 )
         => ( ( finite1152437895449049373et_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_set_nat @ G ) @ C4 ) ) )
     => ( ( finite1152437895449049373et_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_set_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_2728_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_int,C4: nat] :
      ( ! [G: set_int] :
          ( ( ord_less_eq_set_int @ G @ F3 )
         => ( ( finite_finite_int @ G )
           => ( ord_less_eq_nat @ ( finite_card_int @ G ) @ C4 ) ) )
     => ( ( finite_finite_int @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_int @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_2729_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_complex,C4: nat] :
      ( ! [G: set_complex] :
          ( ( ord_le211207098394363844omplex @ G @ F3 )
         => ( ( finite3207457112153483333omplex @ G )
           => ( ord_less_eq_nat @ ( finite_card_complex @ G ) @ C4 ) ) )
     => ( ( finite3207457112153483333omplex @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_complex @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_2730_finite__if__finite__subsets__card__bdd,axiom,
    ! [F3: set_nat,C4: nat] :
      ( ! [G: set_nat] :
          ( ( ord_less_eq_set_nat @ G @ F3 )
         => ( ( finite_finite_nat @ G )
           => ( ord_less_eq_nat @ ( finite_card_nat @ G ) @ C4 ) ) )
     => ( ( finite_finite_nat @ F3 )
        & ( ord_less_eq_nat @ ( finite_card_nat @ F3 ) @ C4 ) ) ) ).

% finite_if_finite_subsets_card_bdd
thf(fact_2731_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_list_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ S3 ) )
     => ~ ! [T3: set_list_nat] :
            ( ( ord_le6045566169113846134st_nat @ T3 @ S3 )
           => ( ( ( finite_card_list_nat @ T3 )
                = N )
             => ~ ( finite8100373058378681591st_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_2732_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ S3 ) )
     => ~ ! [T3: set_set_nat] :
            ( ( ord_le6893508408891458716et_nat @ T3 @ S3 )
           => ( ( ( finite_card_set_nat @ T3 )
                = N )
             => ~ ( finite1152437895449049373et_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_2733_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_int] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_int @ S3 ) )
     => ~ ! [T3: set_int] :
            ( ( ord_less_eq_set_int @ T3 @ S3 )
           => ( ( ( finite_card_int @ T3 )
                = N )
             => ~ ( finite_finite_int @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_2734_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_complex] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_complex @ S3 ) )
     => ~ ! [T3: set_complex] :
            ( ( ord_le211207098394363844omplex @ T3 @ S3 )
           => ( ( ( finite_card_complex @ T3 )
                = N )
             => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_2735_obtain__subset__with__card__n,axiom,
    ! [N: nat,S3: set_nat] :
      ( ( ord_less_eq_nat @ N @ ( finite_card_nat @ S3 ) )
     => ~ ! [T3: set_nat] :
            ( ( ord_less_eq_set_nat @ T3 @ S3 )
           => ( ( ( finite_card_nat @ T3 )
                = N )
             => ~ ( finite_finite_nat @ T3 ) ) ) ) ).

% obtain_subset_with_card_n
thf(fact_2736_power__le__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ one_one_Code_integer ) ) ) ).

% power_le_one
thf(fact_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_power__less__power__Suc,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_less_power_Suc
thf(fact_2750_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_2751_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_2752_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_2753_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_2754_power__gt1__lemma,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_gt1_lemma
thf(fact_2755_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_2756_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_2757_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_2758_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_2759_card__le__sym__Diff,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite8100373058378681591st_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2760_card__le__sym__Diff,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite1152437895449049373et_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2761_card__le__sym__Diff,axiom,
    ! [A4: set_int,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2762_card__le__sym__Diff,axiom,
    ! [A4: set_complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2763_card__le__sym__Diff,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2764_card__le__sym__Diff,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) )
         => ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) ) ) ) ).

% card_le_sym_Diff
thf(fact_2765_card__length,axiom,
    ! [Xs2: list_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( set_complex2 @ Xs2 ) ) @ ( size_s3451745648224563538omplex @ Xs2 ) ) ).

% card_length
thf(fact_2766_card__length,axiom,
    ! [Xs2: list_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) ) @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ).

% card_length
thf(fact_2767_card__length,axiom,
    ! [Xs2: list_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( set_set_nat2 @ Xs2 ) ) @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ).

% card_length
thf(fact_2768_card__length,axiom,
    ! [Xs2: list_VEBT_VEBT] : ( ord_less_eq_nat @ ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% card_length
thf(fact_2769_card__length,axiom,
    ! [Xs2: list_o] : ( ord_less_eq_nat @ ( finite_card_o @ ( set_o2 @ Xs2 ) ) @ ( size_size_list_o @ Xs2 ) ) ).

% card_length
thf(fact_2770_card__length,axiom,
    ! [Xs2: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs2 ) ) @ ( size_size_list_nat @ Xs2 ) ) ).

% card_length
thf(fact_2771_card__length,axiom,
    ! [Xs2: list_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( set_int2 @ Xs2 ) ) @ ( size_size_list_int @ Xs2 ) ) ).

% card_length
thf(fact_2772_power__gt1,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) ) ) ).

% power_gt1
thf(fact_2773_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_2774_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_2775_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_2776_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_2777_power__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( power_8256067586552552935nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% power_0_left
thf(fact_2778_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_2779_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_2780_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_2781_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_2782_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_2783_power__increasing,axiom,
    ! [N: nat,N6: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N ) @ ( power_8256067586552552935nteger @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2784_power__increasing,axiom,
    ! [N: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N ) @ ( power_power_real @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2785_power__increasing,axiom,
    ! [N: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N ) @ ( power_power_rat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2786_power__increasing,axiom,
    ! [N: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2787_power__increasing,axiom,
    ! [N: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N6 ) ) ) ) ).

% power_increasing
thf(fact_2788_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_2789_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_2790_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_2791_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_2792_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_2793_lenlex__length,axiom,
    ! [Ms: list_VEBT_VEBT,Ns: list_VEBT_VEBT,R3: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Ms @ Ns ) @ ( lenlex_VEBT_VEBT @ R3 ) )
     => ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Ms ) @ ( size_s6755466524823107622T_VEBT @ Ns ) ) ) ).

% lenlex_length
thf(fact_2794_lenlex__length,axiom,
    ! [Ms: list_o,Ns: list_o,R3: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ms @ Ns ) @ ( lenlex_o @ R3 ) )
     => ( ord_less_eq_nat @ ( size_size_list_o @ Ms ) @ ( size_size_list_o @ Ns ) ) ) ).

% lenlex_length
thf(fact_2795_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_2796_lenlex__length,axiom,
    ! [Ms: list_int,Ns: list_int,R3: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R3 ) )
     => ( ord_less_eq_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) ) ) ).

% lenlex_length
thf(fact_2797_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_2798_nat__one__le__power,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).

% nat_one_le_power
thf(fact_2799_enumerate__step,axiom,
    ! [S3: set_nat,N: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( ord_less_nat @ ( infini8530281810654367211te_nat @ S3 @ N ) @ ( infini8530281810654367211te_nat @ S3 @ ( suc @ N ) ) ) ) ).

% enumerate_step
thf(fact_2800_card__gt__0__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_list_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_list_nat )
        & ( finite8100373058378681591st_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2801_card__gt__0__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_set_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_set_nat )
        & ( finite1152437895449049373et_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2802_card__gt__0__iff,axiom,
    ! [A4: set_complex] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
      = ( ( A4 != bot_bot_set_complex )
        & ( finite3207457112153483333omplex @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2803_card__gt__0__iff,axiom,
    ! [A4: set_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
      = ( ( A4 != bot_bot_set_nat )
        & ( finite_finite_nat @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2804_card__gt__0__iff,axiom,
    ! [A4: set_int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
      = ( ( A4 != bot_bot_set_int )
        & ( finite_finite_int @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2805_card__gt__0__iff,axiom,
    ! [A4: set_real] :
      ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
      = ( ( A4 != bot_bot_set_real )
        & ( finite_finite_real @ A4 ) ) ) ).

% card_gt_0_iff
thf(fact_2806_card__1__singleton__iff,axiom,
    ! [A4: set_complex] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: complex] :
            ( A4
            = ( insert_complex @ X4 @ bot_bot_set_complex ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2807_card__1__singleton__iff,axiom,
    ! [A4: set_list_nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: list_nat] :
            ( A4
            = ( insert_list_nat @ X4 @ bot_bot_set_list_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2808_card__1__singleton__iff,axiom,
    ! [A4: set_set_nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: set_nat] :
            ( A4
            = ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2809_card__1__singleton__iff,axiom,
    ! [A4: set_nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: nat] :
            ( A4
            = ( insert_nat @ X4 @ bot_bot_set_nat ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2810_card__1__singleton__iff,axiom,
    ! [A4: set_int] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: int] :
            ( A4
            = ( insert_int @ X4 @ bot_bot_set_int ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2811_card__1__singleton__iff,axiom,
    ! [A4: set_real] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ zero_zero_nat ) )
      = ( ? [X4: real] :
            ( A4
            = ( insert_real @ X4 @ bot_bot_set_real ) ) ) ) ).

% card_1_singleton_iff
thf(fact_2812_card__eq__SucD,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: complex,B8: set_complex] :
          ( ( A4
            = ( insert_complex @ B3 @ B8 ) )
          & ~ ( member_complex @ B3 @ B8 )
          & ( ( finite_card_complex @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_complex ) ) ) ) ).

% card_eq_SucD
thf(fact_2813_card__eq__SucD,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: list_nat,B8: set_list_nat] :
          ( ( A4
            = ( insert_list_nat @ B3 @ B8 ) )
          & ~ ( member_list_nat @ B3 @ B8 )
          & ( ( finite_card_list_nat @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_list_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2814_card__eq__SucD,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: set_nat,B8: set_set_nat] :
          ( ( A4
            = ( insert_set_nat @ B3 @ B8 ) )
          & ~ ( member_set_nat @ B3 @ B8 )
          & ( ( finite_card_set_nat @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2815_card__eq__SucD,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: nat,B8: set_nat] :
          ( ( A4
            = ( insert_nat @ B3 @ B8 ) )
          & ~ ( member_nat @ B3 @ B8 )
          & ( ( finite_card_nat @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_nat ) ) ) ) ).

% card_eq_SucD
thf(fact_2816_card__eq__SucD,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: int,B8: set_int] :
          ( ( A4
            = ( insert_int @ B3 @ B8 ) )
          & ~ ( member_int @ B3 @ B8 )
          & ( ( finite_card_int @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_int ) ) ) ) ).

% card_eq_SucD
thf(fact_2817_card__eq__SucD,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K2 ) )
     => ? [B3: real,B8: set_real] :
          ( ( A4
            = ( insert_real @ B3 @ B8 ) )
          & ~ ( member_real @ B3 @ B8 )
          & ( ( finite_card_real @ B8 )
            = K2 )
          & ( ( K2 = zero_zero_nat )
           => ( B8 = bot_bot_set_real ) ) ) ) ).

% card_eq_SucD
thf(fact_2818_card__Suc__eq,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( ( finite_card_complex @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: complex,B7: set_complex] :
            ( ( A4
              = ( insert_complex @ B4 @ B7 ) )
            & ~ ( member_complex @ B4 @ B7 )
            & ( ( finite_card_complex @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_complex ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2819_card__Suc__eq,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( ( finite_card_list_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: list_nat,B7: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ B4 @ B7 ) )
            & ~ ( member_list_nat @ B4 @ B7 )
            & ( ( finite_card_list_nat @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_list_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2820_card__Suc__eq,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( ( finite_card_set_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: set_nat,B7: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ B4 @ B7 ) )
            & ~ ( member_set_nat @ B4 @ B7 )
            & ( ( finite_card_set_nat @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2821_card__Suc__eq,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( ( finite_card_nat @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: nat,B7: set_nat] :
            ( ( A4
              = ( insert_nat @ B4 @ B7 ) )
            & ~ ( member_nat @ B4 @ B7 )
            & ( ( finite_card_nat @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_nat ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2822_card__Suc__eq,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( ( finite_card_int @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: int,B7: set_int] :
            ( ( A4
              = ( insert_int @ B4 @ B7 ) )
            & ~ ( member_int @ B4 @ B7 )
            & ( ( finite_card_int @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_int ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2823_card__Suc__eq,axiom,
    ! [A4: set_real,K2: nat] :
      ( ( ( finite_card_real @ A4 )
        = ( suc @ K2 ) )
      = ( ? [B4: real,B7: set_real] :
            ( ( A4
              = ( insert_real @ B4 @ B7 ) )
            & ~ ( member_real @ B4 @ B7 )
            & ( ( finite_card_real @ B7 )
              = K2 )
            & ( ( K2 = zero_zero_nat )
             => ( B7 = bot_bot_set_real ) ) ) ) ) ).

% card_Suc_eq
thf(fact_2824_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_list_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: list_nat] :
              ( ( member_list_nat @ X4 @ A4 )
             => ! [Y4: list_nat] :
                  ( ( member_list_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_2825_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_set_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: set_nat] :
              ( ( member_set_nat @ X4 @ A4 )
             => ! [Y4: set_nat] :
                  ( ( member_set_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_2826_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_2827_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A4 )
             => ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_2828_card__le__Suc0__iff__eq,axiom,
    ! [A4: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ ( suc @ zero_zero_nat ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
             => ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( X4 = Y4 ) ) ) ) ) ) ).

% card_le_Suc0_iff_eq
thf(fact_2829_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_real] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_real @ A4 ) )
      = ( ? [A5: real,B7: set_real] :
            ( ( A4
              = ( insert_real @ A5 @ B7 ) )
            & ~ ( member_real @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_real @ B7 ) )
            & ( finite_finite_real @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2830_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_list_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_list_nat @ A4 ) )
      = ( ? [A5: list_nat,B7: set_list_nat] :
            ( ( A4
              = ( insert_list_nat @ A5 @ B7 ) )
            & ~ ( member_list_nat @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_list_nat @ B7 ) )
            & ( finite8100373058378681591st_nat @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2831_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_set_nat @ A4 ) )
      = ( ? [A5: set_nat,B7: set_set_nat] :
            ( ( A4
              = ( insert_set_nat @ A5 @ B7 ) )
            & ~ ( member_set_nat @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_set_nat @ B7 ) )
            & ( finite1152437895449049373et_nat @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2832_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_nat @ A4 ) )
      = ( ? [A5: nat,B7: set_nat] :
            ( ( A4
              = ( insert_nat @ A5 @ B7 ) )
            & ~ ( member_nat @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_nat @ B7 ) )
            & ( finite_finite_nat @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2833_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_int] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_int @ A4 ) )
      = ( ? [A5: int,B7: set_int] :
            ( ( A4
              = ( insert_int @ A5 @ B7 ) )
            & ~ ( member_int @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_int @ B7 ) )
            & ( finite_finite_int @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2834_card__le__Suc__iff,axiom,
    ! [N: nat,A4: set_complex] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( finite_card_complex @ A4 ) )
      = ( ? [A5: complex,B7: set_complex] :
            ( ( A4
              = ( insert_complex @ A5 @ B7 ) )
            & ~ ( member_complex @ A5 @ B7 )
            & ( ord_less_eq_nat @ N @ ( finite_card_complex @ B7 ) )
            & ( finite3207457112153483333omplex @ B7 ) ) ) ) ).

% card_le_Suc_iff
thf(fact_2835_power__Suc__less,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% power_Suc_less
thf(fact_2836_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_2837_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_2838_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_2839_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_2840_power__Suc__le__self,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2841_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_2842_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_2843_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_2844_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_2845_power__Suc__less__one,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
       => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ ( suc @ N ) ) @ one_one_Code_integer ) ) ) ).

% power_Suc_less_one
thf(fact_2846_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_2847_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_2848_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_2849_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_2850_card__Diff1__le,axiom,
    ! [A4: set_complex,X: complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) @ ( finite_card_complex @ A4 ) ) ).

% card_Diff1_le
thf(fact_2851_card__Diff1__le,axiom,
    ! [A4: set_list_nat,X: list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) @ ( finite_card_list_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_2852_card__Diff1__le,axiom,
    ! [A4: set_set_nat,X: set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) @ ( finite_card_set_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_2853_card__Diff1__le,axiom,
    ! [A4: set_int,X: int] : ( ord_less_eq_nat @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A4 ) ) ).

% card_Diff1_le
thf(fact_2854_card__Diff1__le,axiom,
    ! [A4: set_real,X: real] : ( ord_less_eq_nat @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) @ ( finite_card_real @ A4 ) ) ).

% card_Diff1_le
thf(fact_2855_card__Diff1__le,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] : ( ord_less_eq_nat @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) @ ( finite711546835091564841at_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_2856_card__Diff1__le,axiom,
    ! [A4: set_nat,X: nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A4 ) ) ).

% card_Diff1_le
thf(fact_2857_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A: code_integer] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le6747313008572928689nteger @ A @ one_one_Code_integer )
         => ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ A @ N6 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2858_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A: real] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2859_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A: rat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2860_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A: nat] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2861_power__strict__decreasing,axiom,
    ! [N: nat,N6: nat,A: int] :
      ( ( ord_less_nat @ N @ N6 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2862_power__decreasing,axiom,
    ! [N: nat,N6: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
       => ( ( ord_le3102999989581377725nteger @ A @ one_one_Code_integer )
         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N6 ) @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2863_power__decreasing,axiom,
    ! [N: nat,N6: nat,A: real] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N6 ) @ ( power_power_real @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2864_power__decreasing,axiom,
    ! [N: nat,N6: nat,A: rat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N6 ) @ ( power_power_rat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2865_power__decreasing,axiom,
    ! [N: nat,N6: nat,A: nat] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N6 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2866_power__decreasing,axiom,
    ! [N: nat,N6: nat,A: int] :
      ( ( ord_less_eq_nat @ N @ N6 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N6 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).

% power_decreasing
thf(fact_2867_power__le__imp__le__exp,axiom,
    ! [A: code_integer,M2: nat,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_le_imp_le_exp
thf(fact_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_diff__card__le__card__Diff,axiom,
    ! [B5: set_list_nat,A4: set_list_nat] :
      ( ( finite8100373058378681591st_nat @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2881_diff__card__le__card__Diff,axiom,
    ! [B5: set_set_nat,A4: set_set_nat] :
      ( ( finite1152437895449049373et_nat @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2882_diff__card__le__card__Diff,axiom,
    ! [B5: set_int,A4: set_int] :
      ( ( finite_finite_int @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2883_diff__card__le__card__Diff,axiom,
    ! [B5: set_complex,A4: set_complex] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B5 ) ) @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2884_diff__card__le__card__Diff,axiom,
    ! [B5: set_Pr1261947904930325089at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite711546835091564841at_nat @ A4 ) @ ( finite711546835091564841at_nat @ B5 ) ) @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2885_diff__card__le__card__Diff,axiom,
    ! [B5: set_nat,A4: set_nat] :
      ( ( finite_finite_nat @ B5 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ).

% diff_card_le_card_Diff
thf(fact_2886_self__le__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le3102999989581377725nteger @ A @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% self_le_power
thf(fact_2887_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_2888_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_2889_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_2890_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_2891_one__less__power,axiom,
    ! [A: code_integer,N: nat] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ N ) ) ) ) ).

% one_less_power
thf(fact_2892_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_2893_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_2894_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_2895_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_2896_count__le__length,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs2 @ X ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% count_le_length
thf(fact_2897_count__le__length,axiom,
    ! [Xs2: list_o,X: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs2 @ X ) @ ( size_size_list_o @ Xs2 ) ) ).

% count_le_length
thf(fact_2898_count__le__length,axiom,
    ! [Xs2: list_nat,X: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs2 @ X ) @ ( size_size_list_nat @ Xs2 ) ) ).

% count_le_length
thf(fact_2899_count__le__length,axiom,
    ! [Xs2: list_int,X: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs2 @ X ) @ ( size_size_list_int @ Xs2 ) ) ).

% count_le_length
thf(fact_2900_dbl__inc__def,axiom,
    ( neg_nu5831290666863070958nteger
    = ( ^ [X4: code_integer] : ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ X4 @ X4 ) @ one_one_Code_integer ) ) ) ).

% dbl_inc_def
thf(fact_2901_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X4: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X4 @ X4 ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_2902_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_2903_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_2904_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_2905_card_Oremove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ( finite_card_list_nat @ A4 )
          = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2906_card_Oremove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ( finite_card_set_nat @ A4 )
          = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2907_card_Oremove,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( finite_card_complex @ A4 )
          = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2908_card_Oremove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( finite_card_int @ A4 )
          = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2909_card_Oremove,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( finite_card_real @ A4 )
          = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2910_card_Oremove,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( ( finite711546835091564841at_nat @ A4 )
          = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2911_card_Oremove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( finite_card_nat @ A4 )
          = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% card.remove
thf(fact_2912_card_Oinsert__remove,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite_card_list_nat @ ( insert_list_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2913_card_Oinsert__remove,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite_card_set_nat @ ( insert_set_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2914_card_Oinsert__remove,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite_card_complex @ ( insert_complex @ X @ A4 ) )
        = ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2915_card_Oinsert__remove,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_int @ ( insert_int @ X @ A4 ) )
        = ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2916_card_Oinsert__remove,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite_card_real @ ( insert_real @ X @ A4 ) )
        = ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2917_card_Oinsert__remove,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite711546835091564841at_nat @ ( insert8211810215607154385at_nat @ X @ A4 ) )
        = ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2918_card_Oinsert__remove,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_nat @ ( insert_nat @ X @ A4 ) )
        = ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% card.insert_remove
thf(fact_2919_card__Suc__Diff1,axiom,
    ! [A4: set_list_nat,X: list_nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( member_list_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_list_nat @ ( minus_7954133019191499631st_nat @ A4 @ ( insert_list_nat @ X @ bot_bot_set_list_nat ) ) ) )
          = ( finite_card_list_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2920_card__Suc__Diff1,axiom,
    ! [A4: set_set_nat,X: set_nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( member_set_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_set_nat @ ( minus_2163939370556025621et_nat @ A4 @ ( insert_set_nat @ X @ bot_bot_set_set_nat ) ) ) )
          = ( finite_card_set_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2921_card__Suc__Diff1,axiom,
    ! [A4: set_complex,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( suc @ ( finite_card_complex @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) )
          = ( finite_card_complex @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2922_card__Suc__Diff1,axiom,
    ! [A4: set_int,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
          = ( finite_card_int @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2923_card__Suc__Diff1,axiom,
    ! [A4: set_real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( suc @ ( finite_card_real @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) )
          = ( finite_card_real @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2924_card__Suc__Diff1,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( member8440522571783428010at_nat @ X @ A4 )
       => ( ( suc @ ( finite711546835091564841at_nat @ ( minus_1356011639430497352at_nat @ A4 @ ( insert8211810215607154385at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) )
          = ( finite711546835091564841at_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2925_card__Suc__Diff1,axiom,
    ! [A4: set_nat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
          = ( finite_card_nat @ A4 ) ) ) ) ).

% card_Suc_Diff1
thf(fact_2926_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_2927_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_2928_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_2929_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_2930_power__eq__if,axiom,
    ( power_8256067586552552935nteger
    = ( ^ [P5: code_integer,M6: nat] : ( if_Code_integer @ ( M6 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ P5 @ ( power_8256067586552552935nteger @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2931_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2932_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2933_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2934_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2935_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_2936_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_2937_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_2938_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_2939_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_2940_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_2941_Cons__in__lex,axiom,
    ! [X: code_integer,Xs2: list_Code_integer,Y: code_integer,Ys: list_Code_integer,R3: set_Pr4811707699266497531nteger] :
      ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ ( cons_Code_integer @ X @ Xs2 ) @ ( cons_Code_integer @ Y @ Ys ) ) @ ( lex_Code_integer @ R3 ) )
      = ( ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ R3 )
          & ( ( size_s3445333598471063425nteger @ Xs2 )
            = ( size_s3445333598471063425nteger @ Ys ) ) )
        | ( ( X = Y )
          & ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs2 @ Ys ) @ ( lex_Code_integer @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2942_Cons__in__lex,axiom,
    ! [X: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( lex_Pr8571645452597969515at_nat @ R3 ) )
      = ( ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
          & ( ( size_s5460976970255530739at_nat @ Xs2 )
            = ( size_s5460976970255530739at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( lex_Pr8571645452597969515at_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2943_Cons__in__lex,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Xs2: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( lex_se2245640040323279819at_nat @ R3 ) )
      = ( ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
          & ( ( size_s8736152011456118867at_nat @ Xs2 )
            = ( size_s8736152011456118867at_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs2 @ Ys ) @ ( lex_se2245640040323279819at_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2944_Cons__in__lex,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y: vEBT_VEBT,Ys: list_VEBT_VEBT,R3: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ ( cons_VEBT_VEBT @ Y @ Ys ) ) @ ( lex_VEBT_VEBT @ R3 ) )
      = ( ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X @ Y ) @ R3 )
          & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys ) ) )
        | ( ( X = Y )
          & ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs2 @ Ys ) @ ( lex_VEBT_VEBT @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2945_Cons__in__lex,axiom,
    ! [X: $o,Xs2: list_o,Y: $o,Ys: list_o,R3: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X @ Xs2 ) @ ( cons_o @ Y @ Ys ) ) @ ( lex_o @ R3 ) )
      = ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R3 )
          & ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys ) ) )
        | ( ( X = Y )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys ) @ ( lex_o @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2946_Cons__in__lex,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
          & ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( lex_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2947_Cons__in__lex,axiom,
    ! [X: int,Xs2: list_int,Y: int,Ys: list_int,R3: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R3 ) )
      = ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 )
          & ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys ) ) )
        | ( ( X = Y )
          & ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( lex_int @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_2948_realpow__pos__nth__unique,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X3: real] :
            ( ( ord_less_real @ zero_zero_real @ X3 )
            & ( ( power_power_real @ X3 @ N )
              = A )
            & ! [Y6: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y6 )
                  & ( ( power_power_real @ Y6 @ N )
                    = A ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2949_realpow__pos__nth,axiom,
    ! [N: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ( ( power_power_real @ R4 @ N )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2950_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 )
    & ! [X5: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ( vEBT_invar_vebt @ X5 @ na ) ) ) ).

% sprop1
thf(fact_2951_listrel__iff__nth,axiom,
    ! [Xs2: list_Code_integer,Ys: list_Code_integer,R3: set_Pr4811707699266497531nteger] :
      ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs2 @ Ys ) @ ( listre5734910445319291053nteger @ R3 ) )
      = ( ( ( size_s3445333598471063425nteger @ Xs2 )
          = ( size_s3445333598471063425nteger @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3445333598471063425nteger @ Xs2 ) )
           => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ N4 ) @ ( nth_Code_integer @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2952_listrel__iff__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R3: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs2 @ Ys ) @ ( listre1230615542750757617T_VEBT @ R3 ) )
      = ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N4 ) @ ( nth_VEBT_VEBT @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2953_listrel__iff__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o,R3: set_Pr3175402225741728619VEBT_o] :
      ( ( member3126162362653435956list_o @ ( produc2717590391345394939list_o @ Xs2 @ Ys ) @ ( listrel_VEBT_VEBT_o @ R3 ) )
      = ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_size_list_o @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs2 @ N4 ) @ ( nth_o @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2954_listrel__iff__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,R3: set_Pr7556676689462069481BT_nat] :
      ( ( member6193324644334088288st_nat @ ( produc5570133714943300547st_nat @ Xs2 @ Ys ) @ ( listre5900670229112895443BT_nat @ R3 ) )
      = ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs2 @ N4 ) @ ( nth_nat @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2955_listrel__iff__nth,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int,R3: set_Pr5066593544530342725BT_int] :
      ( ( member3703241499402361532st_int @ ( produc1392282695434103839st_int @ Xs2 @ Ys ) @ ( listre5898179758603845167BT_int @ R3 ) )
      = ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
          = ( size_size_list_int @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
           => ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs2 @ N4 ) @ ( nth_int @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2956_listrel__iff__nth,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT,R3: set_Pr7543698050874017315T_VEBT] :
      ( ( member1087064965665443052T_VEBT @ ( produc6043759678074843571T_VEBT @ Xs2 @ Ys ) @ ( listrel_o_VEBT_VEBT @ R3 ) )
      = ( ( ( size_size_list_o @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_o @ Xs2 ) )
           => ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs2 @ N4 ) @ ( nth_VEBT_VEBT @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2957_listrel__iff__nth,axiom,
    ! [Xs2: list_o,Ys: list_o,R3: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys ) @ ( listrel_o_o @ R3 ) )
      = ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_o @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_o @ Xs2 ) )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs2 @ N4 ) @ ( nth_o @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2958_listrel__iff__nth,axiom,
    ! [Xs2: list_o,Ys: list_nat,R3: set_Pr2101469702781467981_o_nat] :
      ( ( member1519744053835550788st_nat @ ( produc7128876500814652583st_nat @ Xs2 @ Ys ) @ ( listrel_o_nat @ R3 ) )
      = ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_nat @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_o @ Xs2 ) )
           => ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ ( nth_o @ Xs2 @ N4 ) @ ( nth_nat @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2959_listrel__iff__nth,axiom,
    ! [Xs2: list_o,Ys: list_int,R3: set_Pr8834758594704517033_o_int] :
      ( ( member8253032945758599840st_int @ ( produc2951025481305455875st_int @ Xs2 @ Ys ) @ ( listrel_o_int @ R3 ) )
      = ( ( ( size_size_list_o @ Xs2 )
          = ( size_size_list_int @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_o @ Xs2 ) )
           => ( member7847949116333733898_o_int @ ( product_Pair_o_int @ ( nth_o @ Xs2 @ N4 ) @ ( nth_int @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2960_listrel__iff__nth,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT,R3: set_Pr6167073792073659919T_VEBT] :
      ( ( member5968030670617646438T_VEBT @ ( produc8335345208264861441T_VEBT @ Xs2 @ Ys ) @ ( listre5761932458788874033T_VEBT @ R3 ) )
      = ( ( ( size_size_list_nat @ Xs2 )
          = ( size_s6755466524823107622T_VEBT @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs2 ) )
           => ( member8549952807677709168T_VEBT @ ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs2 @ N4 ) @ ( nth_VEBT_VEBT @ Ys @ N4 ) ) @ R3 ) ) ) ) ).

% listrel_iff_nth
thf(fact_2961_in__measures_I2_J,axiom,
    ! [X: code_integer,Y: code_integer,F2: code_integer > nat,Fs: list_C4705013386053401436er_nat] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_2962_in__measures_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,F2: product_prod_nat_nat > nat,Fs: list_P9162950289778280392at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_2963_in__measures_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,F2: set_Pr1261947904930325089at_nat > nat,Fs: list_s9130966667114977576at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ ( cons_s2538900923071588440at_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_2964_in__measures_I2_J,axiom,
    ! [X: nat,Y: nat,F2: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_2965_in__measures_I2_J,axiom,
    ! [X: int,Y: int,F2: int > nat,Fs: list_int_nat] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) )
      = ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
        | ( ( ( F2 @ X )
            = ( F2 @ Y ) )
          & ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_2966_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2967_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2968_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2969_of__nat__zero__less__power__iff,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X )
        | ( N = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_2970__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,Summary3: vEBT_VEBT,Info: option4927543243414619207at_nat] :
        ~ ( ( sa
            = ( vEBT_Node @ Info @ deg @ TreeList3 @ Summary3 ) )
          & ( deg
            = ( plus_plus_nat @ na @ m ) )
          & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
          & ( vEBT_invar_vebt @ Summary3 @ m )
          & ! [X5: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
             => ( vEBT_invar_vebt @ X5 @ 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_2971_intind,axiom,
    ! [I2: nat,N: nat,P2: nat > $o,X: nat] :
      ( ( ord_less_nat @ I2 @ N )
     => ( ( P2 @ X )
       => ( P2 @ ( nth_nat @ ( replicate_nat @ N @ X ) @ I2 ) ) ) ) ).

% intind
thf(fact_2972_intind,axiom,
    ! [I2: nat,N: nat,P2: vEBT_VEBT > $o,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ N )
     => ( ( P2 @ X )
       => ( P2 @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) @ I2 ) ) ) ) ).

% intind
thf(fact_2973_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera1916890842035813515d_enat @ M2 )
        = ( numera1916890842035813515d_enat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_2974_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numera6690914467698888265omplex @ M2 )
        = ( numera6690914467698888265omplex @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_2975_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_real @ M2 )
        = ( numeral_numeral_real @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_2976_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_nat @ M2 )
        = ( numeral_numeral_nat @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_2977_numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( numeral_numeral_int @ M2 )
        = ( numeral_numeral_int @ N ) )
      = ( M2 = N ) ) ).

% numeral_eq_iff
thf(fact_2978_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_2979_semiring__norm_I11_J,axiom,
    ! [M2: num] :
      ( ( times_times_num @ M2 @ one )
      = M2 ) ).

% semiring_norm(11)
thf(fact_2980_semiring__norm_I12_J,axiom,
    ! [N: num] :
      ( ( times_times_num @ one @ N )
      = N ) ).

% semiring_norm(12)
thf(fact_2981_case4_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% case4(10)
thf(fact_2982_case4_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList2 )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% case4(4)
thf(fact_2983_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri5074537144036343181t_real @ M2 )
        = ( semiri5074537144036343181t_real @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_2984_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1314217659103216013at_int @ M2 )
        = ( semiri1314217659103216013at_int @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_2985_of__nat__eq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M2 )
        = ( semiri1316708129612266289at_nat @ N ) )
      = ( M2 = N ) ) ).

% of_nat_eq_iff
thf(fact_2986_a0,axiom,
    ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ).

% a0
thf(fact_2987_case4_I7_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I4 ) @ X7 ) )
        = ( vEBT_V8194947554948674370ptions @ summary2 @ I4 ) ) ) ).

% case4(7)
thf(fact_2988_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X: nat,N: nat] :
      ( ( vEBT_vebt_member @ Tree @ X )
     => ( ( vEBT_invar_vebt @ Tree @ N )
       => ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% member_bound
thf(fact_2989_valid__pres__insert,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_invar_vebt @ ( vEBT_vebt_insert @ T @ X ) @ N ) ) ) ).

% valid_pres_insert
thf(fact_2990_numeral__le__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_eq_num @ M2 @ N ) ) ).

% numeral_le_iff
thf(fact_2991_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_2992_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_2993_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_2994_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_2995_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_2996_numeral__less__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% numeral_less_iff
thf(fact_2997_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_2998_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_2999_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_3000_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_3001_numeral__times__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M2 @ N ) ) ) ).

% numeral_times_numeral
thf(fact_3002_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_3003_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_3004_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_3005_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_3006_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Z3 ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3007_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z3 ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3008_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z3 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3009_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Z3 ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3010_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ W2 ) @ Z3 ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3011_mult__numeral__left__semiring__numeral,axiom,
    ! [V2: num,W2: num,Z3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Z3 ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_3012_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V2 ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3013_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z3 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3014_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W2 ) @ Z3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3015_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3016_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3017_add__numeral__left,axiom,
    ! [V2: num,W2: num,Z3: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V2 @ W2 ) ) @ Z3 ) ) ).

% add_numeral_left
thf(fact_3018_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_3019_numeral__plus__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M2 ) @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M2 @ N ) ) ) ).

% numeral_plus_numeral
thf(fact_3020_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_3021_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_3022_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_3023_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_3024_num__double,axiom,
    ! [N: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N )
      = ( bit0 @ N ) ) ).

% num_double
thf(fact_3025_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_3026_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_3027_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_3028_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_3029_insert__simp__mima,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        | ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% insert_simp_mima
thf(fact_3030_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_3031_not__real__square__gt__zero,axiom,
    ! [X: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
      = ( X = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3032_valid__insert__both__member__options__add,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ X ) @ X ) ) ) ).

% valid_insert_both_member_options_add
thf(fact_3033_valid__insert__both__member__options__pres,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_V8194947554948674370ptions @ T @ X )
           => ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T @ Y ) @ X ) ) ) ) ) ).

% valid_insert_both_member_options_pres
thf(fact_3034_helpypredd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpypredd
thf(fact_3035_helpyd,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = ( some_nat @ Y ) )
       => ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% helpyd
thf(fact_3036_post__member__pre__member,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat,Y: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( ord_less_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
         => ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ T @ X ) @ Y )
           => ( ( vEBT_vebt_member @ T @ Y )
              | ( X = Y ) ) ) ) ) ) ).

% post_member_pre_member
thf(fact_3037_replicate__eq__replicate,axiom,
    ! [M2: nat,X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M2 @ X )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( M2 = N )
        & ( ( M2 != zero_zero_nat )
         => ( X = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_3038_length__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_3039_length__replicate,axiom,
    ! [N: nat,X: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_3040_length__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_3041_length__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( size_size_list_int @ ( replicate_int @ N @ X ) )
      = N ) ).

% length_replicate
thf(fact_3042_delt__out__of__range,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
        | ( ord_less_nat @ Ma @ X ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% delt_out_of_range
thf(fact_3043_del__single__cont,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( X = Ma ) )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) ) ) ) ).

% del_single_cont
thf(fact_3044_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,M2: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
       => ( ord_less_eq_nat @ one_one_nat @ N ) ) ) ).

% set_n_deg_not_0
thf(fact_3045_mi__ma__2__deg,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_eq_nat @ Mi @ Ma )
        & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).

% mi_ma_2_deg
thf(fact_3046_pred__max,axiom,
    ! [Deg: nat,Ma: nat,X: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) ) ) ).

% pred_max
thf(fact_3047_succ__min,axiom,
    ! [Deg: nat,X: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) ) ) ).

% succ_min
thf(fact_3048_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H2: nat,L2: nat,D2: nat] : ( plus_plus_nat @ ( times_times_nat @ H2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D2 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_3049_distrib__left__numeral,axiom,
    ! [V2: num,B: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( plus_plus_rat @ B @ C2 ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3050_distrib__left__numeral,axiom,
    ! [V2: num,B: extended_enat,C2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ ( plus_p3455044024723400733d_enat @ B @ C2 ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3051_distrib__left__numeral,axiom,
    ! [V2: num,B: complex,C2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( plus_plus_complex @ B @ C2 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3052_distrib__left__numeral,axiom,
    ! [V2: num,B: real,C2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( plus_plus_real @ B @ C2 ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3053_distrib__left__numeral,axiom,
    ! [V2: num,B: nat,C2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3054_distrib__left__numeral,axiom,
    ! [V2: num,B: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( plus_plus_int @ B @ C2 ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C2 ) ) ) ).

% distrib_left_numeral
thf(fact_3055_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_3056_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V2: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V2 ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V2 ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V2 ) ) ) ) ).

% distrib_right_numeral
thf(fact_3057_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_3058_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_3059_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_3060_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_3061_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6620942414471956472nteger @ N )
        = one_one_Code_integer )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3062_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera1916890842035813515d_enat @ N )
        = one_on7984719198319812577d_enat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3063_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numera6690914467698888265omplex @ N )
        = one_one_complex )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3064_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_real @ N )
        = one_one_real )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3065_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_nat @ N )
        = one_one_nat )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3066_numeral__eq__one__iff,axiom,
    ! [N: num] :
      ( ( ( numeral_numeral_int @ N )
        = one_one_int )
      = ( N = one ) ) ).

% numeral_eq_one_iff
thf(fact_3067_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_Code_integer
        = ( numera6620942414471956472nteger @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3068_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3069_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3070_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3071_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3072_one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N ) )
      = ( one = N ) ) ).

% one_eq_numeral_iff
thf(fact_3073_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_3074_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_3075_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_3076_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_3077_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: rat,C2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( minus_minus_rat @ B @ C2 ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3078_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: complex,C2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( minus_minus_complex @ B @ C2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3079_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: real,C2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( minus_minus_real @ B @ C2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V2 ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3080_right__diff__distrib__numeral,axiom,
    ! [V2: num,B: int,C2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( minus_minus_int @ B @ C2 ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V2 ) @ C2 ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3081_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_3082_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_3083_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_3084_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_3085_power__zero__numeral,axiom,
    ! [K2: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K2 ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_3086_Suc__numeral,axiom,
    ! [N: num] :
      ( ( suc @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).

% Suc_numeral
thf(fact_3087_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_3088_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_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_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_3096_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_3097_of__nat__eq__0__iff,axiom,
    ! [M2: nat] :
      ( ( ( semiri8010041392384452111omplex @ M2 )
        = zero_zero_complex )
      = ( M2 = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_3098_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_3099_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_3100_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_3101_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_3102_of__nat__0__eq__iff,axiom,
    ! [N: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( zero_zero_nat = N ) ) ).

% of_nat_0_eq_iff
thf(fact_3103_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_3104_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_3105_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_3106_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_3107_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_3108_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_3109_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_3110_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_3111_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_3112_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_3113_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_3114_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_3115_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_3116_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N ) )
      = ( numera1916890842035813515d_enat @ N ) ) ).

% of_nat_numeral
thf(fact_3117_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N ) )
      = ( numera6690914467698888265omplex @ N ) ) ).

% of_nat_numeral
thf(fact_3118_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% of_nat_numeral
thf(fact_3119_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% of_nat_numeral
thf(fact_3120_of__nat__numeral,axiom,
    ! [N: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N ) )
      = ( numeral_numeral_nat @ N ) ) ).

% of_nat_numeral
thf(fact_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri4939895301339042750nteger @ N )
        = one_one_Code_integer )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3134_of__nat__eq__1__iff,axiom,
    ! [N: nat] :
      ( ( ( semiri8010041392384452111omplex @ N )
        = one_one_complex )
      = ( N = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_3135_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_3136_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_3137_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_3138_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_Code_integer
        = ( semiri4939895301339042750nteger @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3139_of__nat__1__eq__iff,axiom,
    ! [N: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N ) )
      = ( N = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_3140_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_3141_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_3142_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_3143_of__nat__1,axiom,
    ( ( semiri4939895301339042750nteger @ one_one_nat )
    = one_one_Code_integer ) ).

% of_nat_1
thf(fact_3144_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_3145_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_3146_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_3147_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_3148_in__set__replicate,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( member_complex @ X @ ( set_complex2 @ ( replicate_complex @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3149_in__set__replicate,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( member_real @ X @ ( set_real2 @ ( replicate_real @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3150_in__set__replicate,axiom,
    ! [X: set_nat,N: nat,Y: set_nat] :
      ( ( member_set_nat @ X @ ( set_set_nat2 @ ( replicate_set_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3151_in__set__replicate,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3152_in__set__replicate,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3153_in__set__replicate,axiom,
    ! [X: vEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ Y ) ) )
      = ( ( X = Y )
        & ( N != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_3154_Bex__set__replicate,axiom,
    ! [N: nat,A: nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
            & ( P2 @ X4 ) ) )
      = ( ( P2 @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3155_Bex__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ? [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
            & ( P2 @ X4 ) ) )
      = ( ( P2 @ A )
        & ( N != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_3156_Ball__set__replicate,axiom,
    ! [N: nat,A: nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N @ A ) ) )
           => ( P2 @ X4 ) ) )
      = ( ( P2 @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3157_Ball__set__replicate,axiom,
    ! [N: nat,A: vEBT_VEBT,P2: vEBT_VEBT > $o] :
      ( ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ A ) ) )
           => ( P2 @ X4 ) ) )
      = ( ( P2 @ A )
        | ( N = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_3158_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3159_numeral__le__one__iff,axiom,
    ! [N: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( ord_less_eq_num @ N @ one ) ) ).

% numeral_le_one_iff
thf(fact_3160_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_3161_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_3162_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_3163_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_3164_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3165_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_3166_one__less__numeral__iff,axiom,
    ! [N: num] :
      ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( ord_less_num @ one @ N ) ) ).

% one_less_numeral_iff
thf(fact_3167_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_3168_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_3169_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_3170_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3171_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_3172_one__plus__numeral,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N ) ) ) ).

% one_plus_numeral
thf(fact_3173_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_3174_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_3175_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_3176_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_3177_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3178_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_3179_numeral__plus__one,axiom,
    ! [N: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N @ one ) ) ) ).

% numeral_plus_one
thf(fact_3180_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_3181_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_3182_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_3183_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_3184_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_3185_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_3186_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_3187_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_3188_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri4939895301339042750nteger @ ( suc @ M2 ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( semiri4939895301339042750nteger @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3189_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M2 ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3190_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M2 ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3191_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M2 ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3192_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3193_of__nat__Suc,axiom,
    ! [M2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).

% of_nat_Suc
thf(fact_3194_one__add__one,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3195_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3196_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3197_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3198_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3199_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3200_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_3201_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_3202_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_3203_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_3204_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_3205_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_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_3213_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3214_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3215_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3216_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_3217_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3218_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3219_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3220_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_3221_set__replicate,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( N != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
        = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_3222_set__replicate,axiom,
    ! [N: nat,X: nat] :
      ( ( N != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
        = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_3223_set__replicate,axiom,
    ! [N: nat,X: int] :
      ( ( N != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N @ X ) )
        = ( insert_int @ X @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_3224_set__replicate,axiom,
    ! [N: nat,X: real] :
      ( ( N != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N @ X ) )
        = ( insert_real @ X @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_3225_power2__eq__iff__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_3226_power2__eq__iff__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_3227_power2__eq__iff__nonneg,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_3228_power2__eq__iff__nonneg,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_3229_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_3230_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_3231_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_3232_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_3233_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_3234_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_3235_sum__power2__eq__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3236_sum__power2__eq__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3237_sum__power2__eq__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_3238_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) @ ( semiri5074537144036343181t_real @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3239_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) @ ( semiri681578069525770553at_rat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3240_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ ( semiri1316708129612266289at_nat @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3241_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N: nat,X: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) @ ( semiri1314217659103216013at_int @ X ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) @ X ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_3242_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3243_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3244_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3245_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X: nat,I2: num,N: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N ) )
      = ( ord_less_eq_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_3246_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_3247_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_3248_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_3249_card__2__iff_H,axiom,
    ! [S3: set_nat] :
      ( ( ( finite_card_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ S3 )
            & ? [Y4: nat] :
                ( ( member_nat @ Y4 @ S3 )
                & ( X4 != Y4 )
                & ! [Z4: nat] :
                    ( ( member_nat @ Z4 @ S3 )
                   => ( ( Z4 = X4 )
                      | ( Z4 = Y4 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_3250_card__2__iff_H,axiom,
    ! [S3: set_complex] :
      ( ( ( finite_card_complex @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: complex] :
            ( ( member_complex @ X4 @ S3 )
            & ? [Y4: complex] :
                ( ( member_complex @ Y4 @ S3 )
                & ( X4 != Y4 )
                & ! [Z4: complex] :
                    ( ( member_complex @ Z4 @ S3 )
                   => ( ( Z4 = X4 )
                      | ( Z4 = Y4 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_3251_card__2__iff_H,axiom,
    ! [S3: set_int] :
      ( ( ( finite_card_int @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: int] :
            ( ( member_int @ X4 @ S3 )
            & ? [Y4: int] :
                ( ( member_int @ Y4 @ S3 )
                & ( X4 != Y4 )
                & ! [Z4: int] :
                    ( ( member_int @ Z4 @ S3 )
                   => ( ( Z4 = X4 )
                      | ( Z4 = Y4 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_3252_card__2__iff_H,axiom,
    ! [S3: set_list_nat] :
      ( ( ( finite_card_list_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: list_nat] :
            ( ( member_list_nat @ X4 @ S3 )
            & ? [Y4: list_nat] :
                ( ( member_list_nat @ Y4 @ S3 )
                & ( X4 != Y4 )
                & ! [Z4: list_nat] :
                    ( ( member_list_nat @ Z4 @ S3 )
                   => ( ( Z4 = X4 )
                      | ( Z4 = Y4 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_3253_card__2__iff_H,axiom,
    ! [S3: set_set_nat] :
      ( ( ( finite_card_set_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ S3 )
            & ? [Y4: set_nat] :
                ( ( member_set_nat @ Y4 @ S3 )
                & ( X4 != Y4 )
                & ! [Z4: set_nat] :
                    ( ( member_set_nat @ Z4 @ S3 )
                   => ( ( Z4 = X4 )
                      | ( Z4 = Y4 ) ) ) ) ) ) ) ).

% card_2_iff'
thf(fact_3254_reals__Archimedean3,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ! [Y6: real] :
        ? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% reals_Archimedean3
thf(fact_3255_add__One__commute,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ N )
      = ( plus_plus_num @ N @ one ) ) ).

% add_One_commute
thf(fact_3256_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_3257_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_Bit0
thf(fact_3258_numeral__Bit0,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_Bit0
thf(fact_3259_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_3260_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_3261_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_3262_mult__numeral__1__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3263_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3264_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3265_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3266_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3267_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3268_mult__numeral__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3269_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3270_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3271_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3272_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3273_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3274_numeral__One,axiom,
    ( ( numera6620942414471956472nteger @ one )
    = one_one_Code_integer ) ).

% numeral_One
thf(fact_3275_numeral__One,axiom,
    ( ( numera1916890842035813515d_enat @ one )
    = one_on7984719198319812577d_enat ) ).

% numeral_One
thf(fact_3276_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_3277_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_3278_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_3279_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_3280_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3281_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_3282_le__num__One__iff,axiom,
    ! [X: num] :
      ( ( ord_less_eq_num @ X @ one )
      = ( X = one ) ) ).

% le_num_One_iff
thf(fact_3283_mult__2,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_rat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3284_mult__2,axiom,
    ! [Z3: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_p3455044024723400733d_enat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3285_mult__2,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_complex @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3286_mult__2,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_real @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3287_mult__2,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3288_mult__2,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z3 )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2
thf(fact_3289_mult__2__right,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ Z3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3290_mult__2__right,axiom,
    ! [Z3: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z3 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3291_mult__2__right,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ Z3 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3292_mult__2__right,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ Z3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3293_mult__2__right,axiom,
    ! [Z3: nat] :
      ( ( times_times_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3294_mult__2__right,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ Z3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z3 @ Z3 ) ) ).

% mult_2_right
thf(fact_3295_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_3296_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3297_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_3298_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_3299_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_3300_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_3301_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_3302_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_3303_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_3304_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_3305_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_3306_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_3307_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_3308_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_3309_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_3310_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_3311_power4__eq__xxxx,axiom,
    ! [X: complex] :
      ( ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3312_power4__eq__xxxx,axiom,
    ! [X: real] :
      ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3313_power4__eq__xxxx,axiom,
    ! [X: rat] :
      ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_rat @ ( times_times_rat @ ( times_times_rat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3314_power4__eq__xxxx,axiom,
    ! [X: nat] :
      ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3315_power4__eq__xxxx,axiom,
    ! [X: int] :
      ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X @ X ) @ X ) @ X ) ) ).

% power4_eq_xxxx
thf(fact_3316_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_power2__sum,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3324_power2__sum,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3325_power2__sum,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3326_power2__sum,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3327_power2__sum,axiom,
    ! [X: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3328_power2__sum,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_sum
thf(fact_3329_sum__squares__bound,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3330_sum__squares__bound,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3331_power2__diff,axiom,
    ! [X: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3332_power2__diff,axiom,
    ! [X: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3333_power2__diff,axiom,
    ! [X: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3334_power2__diff,axiom,
    ! [X: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X ) @ Y ) ) ) ).

% power2_diff
thf(fact_3335_power2__le__imp__le,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3336_power2__le__imp__le,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3337_power2__le__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3338_power2__le__imp__le,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_3339_power2__eq__imp__eq,axiom,
    ! [X: real,Y: real] :
      ( ( ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3340_power2__eq__imp__eq,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3341_power2__eq__imp__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3342_power2__eq__imp__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_3343_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_3344_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_3345_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_3346_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_3347_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_3348_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_3349_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_3350_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_3351_card__2__iff,axiom,
    ! [S3: set_complex] :
      ( ( ( finite_card_complex @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: complex,Y4: complex] :
            ( ( S3
              = ( insert_complex @ X4 @ ( insert_complex @ Y4 @ bot_bot_set_complex ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3352_card__2__iff,axiom,
    ! [S3: set_list_nat] :
      ( ( ( finite_card_list_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: list_nat,Y4: list_nat] :
            ( ( S3
              = ( insert_list_nat @ X4 @ ( insert_list_nat @ Y4 @ bot_bot_set_list_nat ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3353_card__2__iff,axiom,
    ! [S3: set_set_nat] :
      ( ( ( finite_card_set_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: set_nat,Y4: set_nat] :
            ( ( S3
              = ( insert_set_nat @ X4 @ ( insert_set_nat @ Y4 @ bot_bot_set_set_nat ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3354_card__2__iff,axiom,
    ! [S3: set_nat] :
      ( ( ( finite_card_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: nat,Y4: nat] :
            ( ( S3
              = ( insert_nat @ X4 @ ( insert_nat @ Y4 @ bot_bot_set_nat ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3355_card__2__iff,axiom,
    ! [S3: set_int] :
      ( ( ( finite_card_int @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: int,Y4: int] :
            ( ( S3
              = ( insert_int @ X4 @ ( insert_int @ Y4 @ bot_bot_set_int ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3356_card__2__iff,axiom,
    ! [S3: set_real] :
      ( ( ( finite_card_real @ S3 )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( ? [X4: real,Y4: real] :
            ( ( S3
              = ( insert_real @ X4 @ ( insert_real @ Y4 @ bot_bot_set_real ) ) )
            & ( X4 != Y4 ) ) ) ) ).

% card_2_iff
thf(fact_3357_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_3358_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_3359_real__arch__simple,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_3360_real__arch__simple,axiom,
    ! [X: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_3361_reals__Archimedean2,axiom,
    ! [X: rat] :
    ? [N3: nat] : ( ord_less_rat @ X @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_3362_reals__Archimedean2,axiom,
    ! [X: real] :
    ? [N3: nat] : ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_3363_mult__of__nat__commute,axiom,
    ! [X: nat,Y: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X ) @ Y )
      = ( times_times_rat @ Y @ ( semiri681578069525770553at_rat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3364_mult__of__nat__commute,axiom,
    ! [X: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3365_mult__of__nat__commute,axiom,
    ! [X: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3366_mult__of__nat__commute,axiom,
    ! [X: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X ) ) ) ).

% mult_of_nat_commute
thf(fact_3367_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z4: int] :
        ? [N4: nat] :
          ( Z4
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_3368_int__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).

% int_Suc
thf(fact_3369_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_3370_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N ) ) ).

% zero_neq_numeral
thf(fact_3371_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N ) ) ).

% zero_neq_numeral
thf(fact_3372_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N ) ) ).

% zero_neq_numeral
thf(fact_3373_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N ) ) ).

% zero_neq_numeral
thf(fact_3374_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N ) ) ).

% zero_neq_numeral
thf(fact_3375_zero__neq__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N ) ) ).

% zero_neq_numeral
thf(fact_3376_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_3377_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N4: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_3378_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_3379_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_3380_int__ops_I7_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(7)
thf(fact_3381_power2__less__imp__less,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3382_power2__less__imp__less,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3383_power2__less__imp__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3384_power2__less__imp__less,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_3385_sum__power2__le__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3386_sum__power2__le__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3387_sum__power2__le__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_3388_sum__power2__ge__zero,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3389_sum__power2__ge__zero,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3390_sum__power2__ge__zero,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_3391_sum__power2__gt__zero__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3392_sum__power2__gt__zero__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3393_sum__power2__gt__zero__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_3394_not__sum__power2__lt__zero,axiom,
    ! [X: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_3395_not__sum__power2__lt__zero,axiom,
    ! [X: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_3396_not__sum__power2__lt__zero,axiom,
    ! [X: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_3397_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_3398_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_3399_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_3400_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_3401_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_3402_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_3403_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_3404_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_3405_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_3406_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 )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
            & ( ord_less_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_3407_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 )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K2 )
            & ( ord_less_eq_nat @ K2 @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_3408_VEBT__internal_Oinsert_H_Osimps_I2_J,axiom,
    ! [Deg: nat,X: nat,Info2: option4927543243414619207at_nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) )
      & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) @ X )
       => ( ( vEBT_VEBT_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X )
          = ( vEBT_vebt_insert @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% VEBT_internal.insert'.simps(2)
thf(fact_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_VEBT__internal_Oinsert_H_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( vEBT_vebt_insert @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
             => ~ ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
                  & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                   => ( Y
                      = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.elims
thf(fact_3416_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_3417_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_3418_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_3419_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_3420_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M2 ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_3421_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M2 ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_3422_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_3423_of__nat__less__0__iff,axiom,
    ! [M2: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_3424_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_3425_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_3426_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_3427_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_3428_of__nat__neq__0,axiom,
    ! [N: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3438_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3439_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3440_of__nat__mono,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).

% of_nat_mono
thf(fact_3441_zero__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_le_numeral
thf(fact_3442_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_le_numeral
thf(fact_3443_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_le_numeral
thf(fact_3444_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_le_numeral
thf(fact_3445_zero__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_le_numeral
thf(fact_3446_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_3447_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_3448_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_3449_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_3450_not__numeral__le__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_3451_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_3452_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_3453_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_3454_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_3455_not__numeral__less__zero,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_3456_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N ) ) ).

% zero_less_numeral
thf(fact_3457_zero__less__numeral,axiom,
    ! [N: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% zero_less_numeral
thf(fact_3458_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N ) ) ).

% zero_less_numeral
thf(fact_3459_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).

% zero_less_numeral
thf(fact_3460_zero__less__numeral,axiom,
    ! [N: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N ) ) ).

% zero_less_numeral
thf(fact_3461_replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( replicate_nat @ ( suc @ N ) @ X )
      = ( cons_nat @ X @ ( replicate_nat @ N @ X ) ) ) ).

% replicate_Suc
thf(fact_3462_replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( replicate_VEBT_VEBT @ ( suc @ N ) @ X )
      = ( cons_VEBT_VEBT @ X @ ( replicate_VEBT_VEBT @ N @ X ) ) ) ).

% replicate_Suc
thf(fact_3463_real__archimedian__rdiv__eq__0,axiom,
    ! [X: real,C2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ! [M: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ X ) @ C2 ) )
         => ( X = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_3464_one__le__numeral,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ N ) ) ).

% one_le_numeral
thf(fact_3465_one__le__numeral,axiom,
    ! [N: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N ) ) ).

% one_le_numeral
thf(fact_3466_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N ) ) ).

% one_le_numeral
thf(fact_3467_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N ) ) ).

% one_le_numeral
thf(fact_3468_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).

% one_le_numeral
thf(fact_3469_one__le__numeral,axiom,
    ! [N: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N ) ) ).

% one_le_numeral
thf(fact_3470_pos__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ~ ! [N3: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% pos_int_cases
thf(fact_3471_zero__less__imp__eq__int,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( K2
            = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% zero_less_imp_eq_int
thf(fact_3472_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J2: int,K2: nat] :
      ( ( ord_less_int @ I2 @ J2 )
     => ( ( ord_less_nat @ zero_zero_nat @ K2 )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K2 ) @ J2 ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_3473_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ N ) @ one_one_Code_integer ) ).

% not_numeral_less_one
thf(fact_3474_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_3475_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_3476_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_3477_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_3478_not__numeral__less__one,axiom,
    ! [N: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_3479_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer ) ) ).

% one_plus_numeral_commute
thf(fact_3480_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_3481_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_3482_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_3483_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_3484_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_3485_one__plus__numeral__commute,axiom,
    ! [X: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_3486_replicate__length__same,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3487_replicate__length__same,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ! [X3: $o] :
          ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3488_replicate__length__same,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3489_replicate__length__same,axiom,
    ! [Xs2: list_int,X: int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ ( set_int2 @ Xs2 ) )
         => ( X3 = X ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs2 ) @ X )
        = Xs2 ) ) ).

% replicate_length_same
thf(fact_3490_replicate__eqI,axiom,
    ! [Xs2: list_complex,N: nat,X: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs2 )
        = N )
     => ( ! [Y3: complex] :
            ( ( member_complex @ Y3 @ ( set_complex2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_complex @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3491_replicate__eqI,axiom,
    ! [Xs2: list_real,N: nat,X: real] :
      ( ( ( size_size_list_real @ Xs2 )
        = N )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_real @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3492_replicate__eqI,axiom,
    ! [Xs2: list_set_nat,N: nat,X: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs2 )
        = N )
     => ( ! [Y3: set_nat] :
            ( ( member_set_nat @ Y3 @ ( set_set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_set_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3493_replicate__eqI,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = N )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_VEBT_VEBT @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3494_replicate__eqI,axiom,
    ! [Xs2: list_o,N: nat,X: $o] :
      ( ( ( size_size_list_o @ Xs2 )
        = N )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_o @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3495_replicate__eqI,axiom,
    ! [Xs2: list_nat,N: nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs2 )
        = N )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_nat @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3496_replicate__eqI,axiom,
    ! [Xs2: list_int,N: nat,X: int] :
      ( ( ( size_size_list_int @ Xs2 )
        = N )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs2 ) )
           => ( Y3 = X ) )
       => ( Xs2
          = ( replicate_int @ N @ X ) ) ) ) ).

% replicate_eqI
thf(fact_3497_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_3498_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 )
               => ( ! [X3: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_3499_listrel__eq__len,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R3: set_Pr6192946355708809607T_VEBT] :
      ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs2 @ Ys ) @ ( listre1230615542750757617T_VEBT @ R3 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3500_listrel__eq__len,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_o,R3: set_Pr3175402225741728619VEBT_o] :
      ( ( member3126162362653435956list_o @ ( produc2717590391345394939list_o @ Xs2 @ Ys ) @ ( listrel_VEBT_VEBT_o @ R3 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3501_listrel__eq__len,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_nat,R3: set_Pr7556676689462069481BT_nat] :
      ( ( member6193324644334088288st_nat @ ( produc5570133714943300547st_nat @ Xs2 @ Ys ) @ ( listre5900670229112895443BT_nat @ R3 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3502_listrel__eq__len,axiom,
    ! [Xs2: list_VEBT_VEBT,Ys: list_int,R3: set_Pr5066593544530342725BT_int] :
      ( ( member3703241499402361532st_int @ ( produc1392282695434103839st_int @ Xs2 @ Ys ) @ ( listre5898179758603845167BT_int @ R3 ) )
     => ( ( size_s6755466524823107622T_VEBT @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3503_listrel__eq__len,axiom,
    ! [Xs2: list_o,Ys: list_VEBT_VEBT,R3: set_Pr7543698050874017315T_VEBT] :
      ( ( member1087064965665443052T_VEBT @ ( produc6043759678074843571T_VEBT @ Xs2 @ Ys ) @ ( listrel_o_VEBT_VEBT @ R3 ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3504_listrel__eq__len,axiom,
    ! [Xs2: list_o,Ys: list_o,R3: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys ) @ ( listrel_o_o @ R3 ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3505_listrel__eq__len,axiom,
    ! [Xs2: list_o,Ys: list_nat,R3: set_Pr2101469702781467981_o_nat] :
      ( ( member1519744053835550788st_nat @ ( produc7128876500814652583st_nat @ Xs2 @ Ys ) @ ( listrel_o_nat @ R3 ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3506_listrel__eq__len,axiom,
    ! [Xs2: list_o,Ys: list_int,R3: set_Pr8834758594704517033_o_int] :
      ( ( member8253032945758599840st_int @ ( produc2951025481305455875st_int @ Xs2 @ Ys ) @ ( listrel_o_int @ R3 ) )
     => ( ( size_size_list_o @ Xs2 )
        = ( size_size_list_int @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3507_listrel__eq__len,axiom,
    ! [Xs2: list_nat,Ys: list_VEBT_VEBT,R3: set_Pr6167073792073659919T_VEBT] :
      ( ( member5968030670617646438T_VEBT @ ( produc8335345208264861441T_VEBT @ Xs2 @ Ys ) @ ( listre5761932458788874033T_VEBT @ R3 ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3508_listrel__eq__len,axiom,
    ! [Xs2: list_nat,Ys: list_o,R3: set_Pr3149072824959771635_nat_o] :
      ( ( member6688923169008879818list_o @ ( produc699922362453767013list_o @ Xs2 @ Ys ) @ ( listrel_nat_o @ R3 ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_size_list_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_3509_ex__less__of__nat__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3510_ex__less__of__nat__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X ) ) ) ).

% ex_less_of_nat_mult
thf(fact_3511_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_3512_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_3513_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_3514_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_3515_zdiff__int__split,axiom,
    ! [P2: int > $o,X: nat,Y: nat] :
      ( ( P2 @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X )
         => ( P2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X @ Y )
         => ( P2 @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_3516_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_3517_realpow__pos__nth2,axiom,
    ! [A: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R4: real] :
          ( ( ord_less_real @ zero_zero_real @ R4 )
          & ( ( power_power_real @ R4 @ ( suc @ N ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3518_listrel__Cons2,axiom,
    ! [Xs2: list_Code_integer,Y: code_integer,Ys: list_Code_integer,R3: set_Pr4811707699266497531nteger] :
      ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs2 @ ( cons_Code_integer @ Y @ Ys ) ) @ ( listre5734910445319291053nteger @ R3 ) )
     => ~ ! [X3: code_integer,Xs3: list_Code_integer] :
            ( ( Xs2
              = ( cons_Code_integer @ X3 @ Xs3 ) )
           => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X3 @ Y ) @ R3 )
             => ~ ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs3 @ Ys ) @ ( listre5734910445319291053nteger @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_3519_listrel__Cons2,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R3 ) )
     => ~ ! [X3: product_prod_nat_nat,Xs3: list_P6011104703257516679at_nat] :
            ( ( Xs2
              = ( cons_P6512896166579812791at_nat @ X3 @ Xs3 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ R3 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs3 @ Ys ) @ ( listre818007680106770737at_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_3520_listrel__Cons2,axiom,
    ! [Xs2: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs2 @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre2047417242196832561at_nat @ R3 ) )
     => ~ ! [X3: set_Pr1261947904930325089at_nat,Xs3: list_s1210847774152347623at_nat] :
            ( ( Xs2
              = ( cons_s6881495754146722583at_nat @ X3 @ Xs3 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X3 @ Y ) @ R3 )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs3 @ Ys ) @ ( listre2047417242196832561at_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_3521_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X3 @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_3522_listrel__Cons2,axiom,
    ! [Xs2: list_int,Y: int,Ys: list_int,R3: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R3 ) )
     => ~ ! [X3: int,Xs3: list_int] :
            ( ( Xs2
              = ( cons_int @ X3 @ Xs3 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R3 )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys ) @ ( listrel_int_int @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_3523_listrel__Cons1,axiom,
    ! [Y: code_integer,Ys: list_Code_integer,Xs2: list_Code_integer,R3: set_Pr4811707699266497531nteger] :
      ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ ( cons_Code_integer @ Y @ Ys ) @ Xs2 ) @ ( listre5734910445319291053nteger @ R3 ) )
     => ~ ! [Y3: code_integer,Ys5: list_Code_integer] :
            ( ( Xs2
              = ( cons_Code_integer @ Y3 @ Ys5 ) )
           => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ Y @ Y3 ) @ R3 )
             => ~ ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Ys @ Ys5 ) @ ( listre5734910445319291053nteger @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_3524_listrel__Cons1,axiom,
    ! [Y: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Xs2: list_P6011104703257516679at_nat,R3: set_Pr8693737435421807431at_nat] :
      ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) @ Xs2 ) @ ( listre818007680106770737at_nat @ R3 ) )
     => ~ ! [Y3: product_prod_nat_nat,Ys5: list_P6011104703257516679at_nat] :
            ( ( Xs2
              = ( cons_P6512896166579812791at_nat @ Y3 @ Ys5 ) )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Ys5 ) @ ( listre818007680106770737at_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_3525_listrel__Cons1,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Xs2: list_s1210847774152347623at_nat,R3: set_Pr4329608150637261639at_nat] :
      ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) @ Xs2 ) @ ( listre2047417242196832561at_nat @ R3 ) )
     => ~ ! [Y3: set_Pr1261947904930325089at_nat,Ys5: list_s1210847774152347623at_nat] :
            ( ( Xs2
              = ( cons_s6881495754146722583at_nat @ Y3 @ Ys5 ) )
           => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Ys @ Ys5 ) @ ( listre2047417242196832561at_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_3526_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs2: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs2 ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [Y3: nat,Ys5: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y3 @ Ys5 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_3527_listrel__Cons1,axiom,
    ! [Y: int,Ys: list_int,Xs2: list_int,R3: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs2 ) @ ( listrel_int_int @ R3 ) )
     => ~ ! [Y3: int,Ys5: list_int] :
            ( ( Xs2
              = ( cons_int @ Y3 @ Ys5 ) )
           => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y3 ) @ R3 )
             => ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys5 ) @ ( listrel_int_int @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_3528_listrel_OCons,axiom,
    ! [X: code_integer,Y: code_integer,R3: set_Pr4811707699266497531nteger,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ R3 )
     => ( ( member749217712838834276nteger @ ( produc750622340256944499nteger @ Xs2 @ Ys ) @ ( listre5734910445319291053nteger @ R3 ) )
       => ( member749217712838834276nteger @ ( produc750622340256944499nteger @ ( cons_Code_integer @ X @ Xs2 ) @ ( cons_Code_integer @ Y @ Ys ) ) @ ( listre5734910445319291053nteger @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_3529_listrel_OCons,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,R3: set_Pr8693737435421807431at_nat,Xs2: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ R3 )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs2 @ Ys ) @ ( listre818007680106770737at_nat @ R3 ) )
       => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs2 ) @ ( cons_P6512896166579812791at_nat @ Y @ Ys ) ) @ ( listre818007680106770737at_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_3530_listrel_OCons,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,R3: set_Pr4329608150637261639at_nat,Xs2: list_s1210847774152347623at_nat,Ys: list_s1210847774152347623at_nat] :
      ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ R3 )
     => ( ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ Xs2 @ Ys ) @ ( listre2047417242196832561at_nat @ R3 ) )
       => ( member4080735728053443344at_nat @ ( produc7536900900485677911at_nat @ ( cons_s6881495754146722583at_nat @ X @ Xs2 ) @ ( cons_s6881495754146722583at_nat @ Y @ Ys ) ) @ ( listre2047417242196832561at_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_3531_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat @ R3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs2 ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_3532_listrel_OCons,axiom,
    ! [X: int,Y: int,R3: set_Pr958786334691620121nt_int,Xs2: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R3 )
     => ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R3 ) )
       => ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs2 ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_3533_measures__less,axiom,
    ! [F2: code_integer > nat,X: code_integer,Y: code_integer,Fs: list_C4705013386053401436er_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_3534_measures__less,axiom,
    ! [F2: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_3535_measures__less,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Fs: list_s9130966667114977576at_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ ( cons_s2538900923071588440at_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_3536_measures__less,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_3537_measures__less,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ).

% measures_less
thf(fact_3538_measures__lesseq,axiom,
    ! [F2: code_integer > nat,X: code_integer,Y: code_integer,Fs: list_C4705013386053401436er_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ Fs ) )
       => ( member157494554546826820nteger @ ( produc1086072967326762835nteger @ X @ Y ) @ ( measur8870801148506250077nteger @ ( cons_C1897838848541180310er_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_3539_measures__lesseq,axiom,
    ! [F2: product_prod_nat_nat > nat,X: product_prod_nat_nat,Y: product_prod_nat_nat,Fs: list_P9162950289778280392at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ Fs ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X @ Y ) @ ( measur2679027848233739777at_nat @ ( cons_P4861729644591583992at_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_3540_measures__lesseq,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Fs: list_s9130966667114977576at_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ Fs ) )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X @ Y ) @ ( measur2694323259624372065at_nat @ ( cons_s2538900923071588440at_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_3541_measures__lesseq,axiom,
    ! [F2: nat > nat,X: nat,Y: nat,Fs: list_nat_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ Fs ) )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( measures_nat @ ( cons_nat_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_3542_measures__lesseq,axiom,
    ! [F2: int > nat,X: int,Y: int,Fs: list_int_nat] :
      ( ( ord_less_eq_nat @ ( F2 @ X ) @ ( F2 @ Y ) )
     => ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F2 @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_3543_set__replicate__Suc,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N ) @ X ) )
      = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_3544_set__replicate__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N ) @ X ) )
      = ( insert_nat @ X @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_3545_set__replicate__Suc,axiom,
    ! [N: nat,X: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N ) @ X ) )
      = ( insert_int @ X @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_3546_set__replicate__Suc,axiom,
    ! [N: nat,X: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N ) @ X ) )
      = ( insert_real @ X @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_3547_set__replicate__conv__if,axiom,
    ! [N: nat,X: vEBT_VEBT] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3548_set__replicate__conv__if,axiom,
    ! [N: nat,X: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = bot_bot_set_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N @ X ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3549_set__replicate__conv__if,axiom,
    ! [N: nat,X: int] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = bot_bot_set_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N @ X ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3550_set__replicate__conv__if,axiom,
    ! [N: nat,X: real] :
      ( ( ( N = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = bot_bot_set_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N @ X ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_3551_Cons__replicate__eq,axiom,
    ! [X: nat,Xs2: list_nat,N: nat,Y: nat] :
      ( ( ( cons_nat @ X @ Xs2 )
        = ( replicate_nat @ N @ Y ) )
      = ( ( X = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).

% Cons_replicate_eq
thf(fact_3552_Cons__replicate__eq,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,N: nat,Y: vEBT_VEBT] :
      ( ( ( cons_VEBT_VEBT @ X @ Xs2 )
        = ( replicate_VEBT_VEBT @ N @ Y ) )
      = ( ( X = Y )
        & ( ord_less_nat @ zero_zero_nat @ N )
        & ( Xs2
          = ( replicate_VEBT_VEBT @ ( minus_minus_nat @ N @ one_one_nat ) @ X ) ) ) ) ).

% Cons_replicate_eq
thf(fact_3553_divmod__step__eq,axiom,
    ! [L: num,R3: nat,Q2: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R3 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3554_divmod__step__eq,axiom,
    ! [L: num,R3: int,Q2: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3555_divmod__step__eq,axiom,
    ! [L: num,R3: code_integer,Q2: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q2 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q2 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_3556_mintlistlength,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( Mi != Ma )
       => ( ( ord_less_nat @ Mi @ Ma )
          & ? [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_3557_insert__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_correct
thf(fact_3558_insert__corr,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
       => ( ( sup_sup_set_nat @ ( vEBT_VEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_insert @ T @ X ) ) ) ) ) ).

% insert_corr
thf(fact_3559_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_3560_nat__bit__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( P2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
       => ( ! [N3: nat] :
              ( ( P2 @ N3 )
             => ( P2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_bit_induct
thf(fact_3561_nat__induct2,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ( P2 @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( P2 @ N3 )
             => ( P2 @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct2
thf(fact_3562_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_3563_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_3564_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_3565_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_3566_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_3567_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_3568_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_3569_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_3570_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_3571_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_3572_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_3573_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_3574_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_3575_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_3576_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_3577_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_3578_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_3579_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_3580_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_3581_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_3582_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_3583_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_3584_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_3585_divide__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C2 )
        = ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3586_divide__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ( divide_divide_real @ A @ C2 )
        = ( divide_divide_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3587_divide__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C2 )
        = ( divide_divide_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_3588_divide__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C2 @ A )
        = ( divide1717551699836669952omplex @ C2 @ B ) )
      = ( ( C2 = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3589_divide__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C2 @ A )
        = ( divide_divide_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3590_divide__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C2 @ A )
        = ( divide_divide_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_3591_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_3592_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_3593_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_3594_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_3595_times__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C2 ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_3596_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C2 ) ) ).

% times_divide_eq_right
thf(fact_3597_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3598_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C2 ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3599_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_3600_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C2 )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_3601_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_3602_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C2 )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% divide_divide_eq_left
thf(fact_3603_times__divide__eq__left,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C2 ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_3604_times__divide__eq__left,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C2 ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_3605_times__divide__eq__left,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C2 ) ) ).

% times_divide_eq_left
thf(fact_3606_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_3607_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_3608_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_3609_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_3610_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_3611_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_3612_atLeastAtMost__iff,axiom,
    ! [I2: set_nat,L: set_nat,U: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4548717258645045905et_nat @ L @ U ) )
      = ( ( ord_less_eq_set_nat @ L @ I2 )
        & ( ord_less_eq_set_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3613_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I2 )
        & ( ord_less_eq_rat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3614_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I2 )
        & ( ord_less_eq_num @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3615_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_eq_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3616_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I2 )
        & ( ord_less_eq_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3617_atLeastAtMost__iff,axiom,
    ! [I2: real,L: real,U: real] :
      ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I2 )
        & ( ord_less_eq_real @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_3618_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_3619_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_3620_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_3621_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_3622_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_3623_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_3624_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_3625_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3626_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3627_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3628_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3629_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_3630_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3631_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3632_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3633_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3634_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_3635_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( ( C2 = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
          = zero_zero_complex ) )
      & ( ( C2 != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3636_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ( C2 = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
          = zero_zero_real ) )
      & ( ( C2 != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3637_mult__divide__mult__cancel__left__if,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ( C2 = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
          = zero_zero_rat ) )
      & ( ( C2 != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_3638_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3639_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3640_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_3641_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ B @ C2 ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3642_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ B @ C2 ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3643_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ B @ C2 ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_3644_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3645_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3646_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_3647_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ C2 @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3648_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ C2 @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3649_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ C2 @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_3650_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_3651_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_3652_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_3653_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_3654_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_3655_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_3656_zero__eq__1__divide__iff,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( divide_divide_real @ one_one_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% zero_eq_1_divide_iff
thf(fact_3657_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_3658_one__divide__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( divide_divide_real @ one_one_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% one_divide_eq_0_iff
thf(fact_3659_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_3660_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_3661_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_3662_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_3663_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_3664_divide__self__if,axiom,
    ! [A: complex] :
      ( ( ( A = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = zero_zero_complex ) )
      & ( ( A != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ A @ A )
          = one_one_complex ) ) ) ).

% divide_self_if
thf(fact_3665_divide__self__if,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( divide_divide_real @ A @ A )
          = one_one_real ) ) ) ).

% divide_self_if
thf(fact_3666_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_3667_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_3668_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_3669_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_3670_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_3671_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_3672_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_3673_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_3674_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_3675_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_3676_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_3677_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_3678_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_3679_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_3680_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_3681_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_3682_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_3683_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_3684_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_3685_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_3686_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_3687_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_3688_atLeastatMost__subset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C2 @ D ) )
      = ( ~ ( ord_less_eq_set_nat @ A @ B )
        | ( ( ord_less_eq_set_nat @ C2 @ A )
          & ( ord_less_eq_set_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3689_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C2 @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3690_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C2 @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3691_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C2 @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3692_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C2 @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3693_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C2 @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_3694_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_3695_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_3696_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_3697_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_3698_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_3699_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_3700_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_3701_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_3702_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_3703_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_3704_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C2 @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3705_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C2 @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3706_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C2 @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C2 ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_3707_nat__mult__div__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ( K2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
          = zero_zero_nat ) )
      & ( ( K2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
          = ( divide_divide_nat @ M2 @ N ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_3708_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_3709_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_3710_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_3711_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_3712_zero__less__divide__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_3713_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_3714_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_3715_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_3716_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_3717_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_3718_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_3719_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_3720_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_3721_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_3722_divide__less__0__1__iff,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% divide_less_0_1_iff
thf(fact_3723_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_3724_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_3725_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_3726_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_3727_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_3728_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3729_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3730_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_3731_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3732_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3733_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_3734_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_3735_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_3736_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_3737_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_3738_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_3739_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_3740_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_3741_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_3742_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_3743_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_3744_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_3745_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_3746_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_3747_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_3748_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_3749_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_3750_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_3751_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_3752_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_3753_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_3754_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_3755_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_3756_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_3757_decr__mult__lemma,axiom,
    ! [D: int,P2: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( minus_minus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X5: int] :
              ( ( P2 @ X5 )
             => ( P2 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_3758_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K: int] :
            ( ( P1 @ X3 )
            = ( P1 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ X3 @ Z5 )
             => ( ( P2 @ X3 )
                = ( P1 @ X3 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_3759_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K: int] :
            ( ( P6 @ X3 )
            = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
       => ( ? [Z5: int] :
            ! [X3: int] :
              ( ( ord_less_int @ Z5 @ X3 )
             => ( ( P2 @ X3 )
                = ( P6 @ X3 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_3760_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3761_times__int__code_I1_J,axiom,
    ! [K2: int] :
      ( ( times_times_int @ K2 @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3762_pos__zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ( times_times_int @ M2 @ N )
          = one_one_int )
        = ( ( M2 = one_one_int )
          & ( N = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3763_zmult__zless__mono2,axiom,
    ! [I2: int,J2: int,K2: int] :
      ( ( ord_less_int @ I2 @ J2 )
     => ( ( ord_less_int @ zero_zero_int @ K2 )
       => ( ord_less_int @ ( times_times_int @ K2 @ I2 ) @ ( times_times_int @ K2 @ J2 ) ) ) ) ).

% zmult_zless_mono2
thf(fact_3764_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R5: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( ord_less_int @ B @ R5 )
           => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3765_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R5: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R5 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
       => ( ( ord_less_int @ R5 @ B )
         => ( ( ord_less_int @ R3 @ B )
           => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3766_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q2: int,R3: int,B2: int,Q4: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R5 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R5 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R5 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q4 @ Q2 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3767_zdiv__mono2__lemma,axiom,
    ! [B: int,Q2: int,R3: int,B2: int,Q4: int,R5: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R5 ) )
       => ( ( ord_less_int @ R5 @ B2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B2 )
             => ( ( ord_less_eq_int @ B2 @ B )
               => ( ord_less_eq_int @ Q2 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3768_q__pos__lemma,axiom,
    ! [B2: int,Q4: int,R5: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B2 @ Q4 ) @ R5 ) )
     => ( ( ord_less_int @ R5 @ B2 )
       => ( ( ord_less_int @ zero_zero_int @ B2 )
         => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).

% q_pos_lemma
thf(fact_3769_incr__mult__lemma,axiom,
    ! [D: int,P2: int > $o,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int] :
            ( ( P2 @ X3 )
           => ( P2 @ ( plus_plus_int @ X3 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K2 )
         => ! [X5: int] :
              ( ( P2 @ X5 )
             => ( P2 @ ( plus_plus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3770_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_3771_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_3772_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_3773_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_3774_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_3775_int__distrib_I2_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
      = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(2)
thf(fact_3776_int__distrib_I1_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
      = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(1)
thf(fact_3777_int__distrib_I3_J,axiom,
    ! [Z1: int,Z22: int,W2: int] :
      ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
      = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).

% int_distrib(3)
thf(fact_3778_int__distrib_I4_J,axiom,
    ! [W2: int,Z1: int,Z22: int] :
      ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
      = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).

% int_distrib(4)
thf(fact_3779_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C2 )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C2 @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3780_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C2 )
      = ( divide_divide_real @ A @ ( times_times_real @ C2 @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3781_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C2 )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C2 @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_3782_divide__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z3: complex,W2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z3 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ W2 ) @ ( times_times_complex @ Y @ Z3 ) ) ) ).

% divide_divide_times_eq
thf(fact_3783_divide__divide__times__eq,axiom,
    ! [X: real,Y: real,Z3: real,W2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ W2 ) @ ( times_times_real @ Y @ Z3 ) ) ) ).

% divide_divide_times_eq
thf(fact_3784_divide__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z3: rat,W2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z3 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ W2 ) @ ( times_times_rat @ Y @ Z3 ) ) ) ).

% divide_divide_times_eq
thf(fact_3785_times__divide__times__eq,axiom,
    ! [X: complex,Y: complex,Z3: complex,W2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ Z3 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3786_times__divide__times__eq,axiom,
    ! [X: real,Y: real,Z3: real,W2: real] :
      ( ( times_times_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ Z3 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3787_times__divide__times__eq,axiom,
    ! [X: rat,Y: rat,Z3: rat,W2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ Z3 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ Y @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_3788_atLeastatMost__psubset__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat,D: set_nat] :
      ( ( ord_less_set_set_nat @ ( set_or4548717258645045905et_nat @ A @ B ) @ ( set_or4548717258645045905et_nat @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_set_nat @ A @ B )
          | ( ( ord_less_eq_set_nat @ C2 @ A )
            & ( ord_less_eq_set_nat @ B @ D )
            & ( ( ord_less_set_nat @ C2 @ A )
              | ( ord_less_set_nat @ B @ D ) ) ) )
        & ( ord_less_eq_set_nat @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3789_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C2 @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C2 @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3790_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C2: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C2 @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C2 @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3791_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C2 @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C2 @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3792_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C2 @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C2 @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3793_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C2 @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C2 @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C2 @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C2 @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_3794_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3795_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_3796_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_3797_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_3798_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_3799_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( divide_divide_real @ A @ C2 ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3800_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( divide_divide_rat @ A @ C2 ) ) ) ) ).

% divide_right_mono_neg
thf(fact_3801_divide__nonpos__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_3802_divide__nonpos__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_3803_divide__nonpos__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_3804_divide__nonpos__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_3805_divide__nonneg__nonpos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_3806_divide__nonneg__nonpos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_3807_divide__nonneg__nonneg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_3808_divide__nonneg__nonneg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_3809_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_3810_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_3811_divide__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).

% divide_right_mono
thf(fact_3812_divide__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_right_mono
thf(fact_3813_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_3814_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_3815_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_3816_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_3817_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono
thf(fact_3818_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% divide_strict_right_mono
thf(fact_3819_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_3820_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_3821_divide__less__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C2 != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_3822_divide__less__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C2 != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_3823_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_3824_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_3825_divide__pos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_3826_divide__pos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_3827_divide__pos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_3828_divide__pos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_3829_divide__neg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_3830_divide__neg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_3831_divide__neg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_3832_divide__neg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_3833_all__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
           => ( P2 @ M6 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less
thf(fact_3834_ex__nat__less,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
            & ( P2 @ M6 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less
thf(fact_3835_frac__eq__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X @ Y )
            = ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( ( times_times_complex @ X @ Z3 )
            = ( times_times_complex @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3836_frac__eq__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ( divide_divide_real @ X @ Y )
            = ( divide_divide_real @ W2 @ Z3 ) )
          = ( ( times_times_real @ X @ Z3 )
            = ( times_times_real @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3837_frac__eq__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X @ Y )
            = ( divide_divide_rat @ W2 @ Z3 ) )
          = ( ( times_times_rat @ X @ Z3 )
            = ( times_times_rat @ W2 @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3838_divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C2 )
        = A )
      = ( ( ( C2 != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_3839_divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ( divide_divide_real @ B @ C2 )
        = A )
      = ( ( ( C2 != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_3840_divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C2 )
        = A )
      = ( ( ( C2 != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_3841_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ A @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_3842_eq__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A
        = ( divide_divide_real @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ A @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_3843_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_3844_divide__eq__imp,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C2 ) )
       => ( ( divide1717551699836669952omplex @ B @ C2 )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3845_divide__eq__imp,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C2 ) )
       => ( ( divide_divide_real @ B @ C2 )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3846_divide__eq__imp,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C2 ) )
       => ( ( divide_divide_rat @ B @ C2 )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3847_eq__divide__imp,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C2 )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_3848_eq__divide__imp,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( times_times_real @ A @ C2 )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_3849_eq__divide__imp,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C2 )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C2 ) ) ) ) ).

% eq_divide_imp
thf(fact_3850_nonzero__divide__eq__eq,axiom,
    ! [C2: complex,B: complex,A: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C2 )
          = A )
        = ( B
          = ( times_times_complex @ A @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3851_nonzero__divide__eq__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( C2 != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C2 )
          = A )
        = ( B
          = ( times_times_real @ A @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3852_nonzero__divide__eq__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C2 )
          = A )
        = ( B
          = ( times_times_rat @ A @ C2 ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3853_nonzero__eq__divide__eq,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C2 ) )
        = ( ( times_times_complex @ A @ C2 )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3854_nonzero__eq__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( C2 != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C2 ) )
        = ( ( times_times_real @ A @ C2 )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3855_nonzero__eq__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( C2 != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C2 ) )
        = ( ( times_times_rat @ A @ C2 )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3856_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_3857_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_3858_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_3859_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_3860_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_3861_divide__numeral__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ ( numeral_numeral_rat @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_3862_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_3863_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_3864_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_3865_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_3866_divide__nonpos__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_3867_divide__nonpos__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_3868_divide__nonpos__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3869_divide__nonpos__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_3870_divide__nonneg__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3871_divide__nonneg__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_3872_divide__nonneg__neg,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_3873_divide__nonneg__neg,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_3874_divide__le__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C2 ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C2 @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3875_divide__le__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C2 @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_3876_frac__less2,axiom,
    ! [X: real,Y: real,W2: real,Z3: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_real @ W2 @ Z3 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_3877_frac__less2,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z3 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_3878_frac__less,axiom,
    ! [X: real,Y: real,W2: real,Z3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z3 )
           => ( ord_less_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_3879_frac__less,axiom,
    ! [X: rat,Y: rat,W2: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( ord_less_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z3 )
           => ( ord_less_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_3880_frac__le,axiom,
    ! [Y: real,X: real,W2: real,Z3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z3 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X @ Z3 ) @ ( divide_divide_real @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_3881_frac__le,axiom,
    ! [Y: rat,X: rat,W2: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z3 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Z3 ) @ ( divide_divide_rat @ Y @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_3882_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3883_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3884_divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3885_divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3886_less__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3887_less__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3888_neg__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3889_neg__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3890_neg__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3891_neg__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3892_pos__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3893_pos__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3894_pos__less__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C2 ) )
        = ( ord_less_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3895_pos__less__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3896_mult__imp__div__pos__less,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X @ ( times_times_real @ Z3 @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3897_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X @ ( times_times_rat @ Z3 @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3898_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z3 @ Y ) @ X )
       => ( ord_less_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3899_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z3 @ Y ) @ X )
       => ( ord_less_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3900_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C2 )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3901_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3902_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C2 @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3903_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3904_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_3905_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_3906_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_3907_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_3908_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C2 )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3909_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C2 )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3910_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C2 )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3911_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3912_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3913_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3914_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3915_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3916_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3917_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3918_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3919_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3920_add__frac__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3921_add__frac__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3922_add__frac__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3923_add__frac__num,axiom,
    ! [Y: complex,X: complex,Z3: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ Z3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3924_add__frac__num,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Y ) @ Z3 )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3925_add__frac__num,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3926_add__num__frac,axiom,
    ! [Y: complex,Z3: complex,X: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ X @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3927_add__num__frac,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z3 @ ( divide_divide_real @ X @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3928_add__num__frac,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Z3 @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3929_add__divide__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_3930_add__divide__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_3931_add__divide__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ X @ ( divide_divide_rat @ Y @ Z3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% add_divide_eq_iff
thf(fact_3932_divide__add__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_3933_divide__add__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_3934_divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X @ Z3 ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_add_eq_iff
thf(fact_3935_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3936_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3937_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = A ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z3 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z3 ) @ B ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_3938_diff__frac__eq,axiom,
    ! [Y: complex,Z3: complex,X: complex,W2: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Y ) @ ( divide1717551699836669952omplex @ W2 @ Z3 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ ( times_times_complex @ W2 @ Y ) ) @ ( times_times_complex @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3939_diff__frac__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3940_diff__frac__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_3941_diff__divide__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ X @ ( divide1717551699836669952omplex @ Y @ Z3 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3942_diff__divide__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ X @ ( divide_divide_real @ Y @ Z3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3943_diff__divide__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( minus_minus_rat @ X @ ( divide_divide_rat @ Y @ Z3 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ Y ) @ Z3 ) ) ) ).

% diff_divide_eq_iff
thf(fact_3944_divide__diff__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X @ Z3 ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3945_divide__diff__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X @ Z3 ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3946_divide__diff__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X @ Z3 ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% divide_diff_eq_iff
thf(fact_3947_card__Un__le,axiom,
    ! [A4: set_complex,B5: set_complex] : ( ord_less_eq_nat @ ( finite_card_complex @ ( sup_sup_set_complex @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_complex @ A4 ) @ ( finite_card_complex @ B5 ) ) ) ).

% card_Un_le
thf(fact_3948_card__Un__le,axiom,
    ! [A4: set_int,B5: set_int] : ( ord_less_eq_nat @ ( finite_card_int @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_int @ A4 ) @ ( finite_card_int @ B5 ) ) ) ).

% card_Un_le
thf(fact_3949_card__Un__le,axiom,
    ! [A4: set_list_nat,B5: set_list_nat] : ( ord_less_eq_nat @ ( finite_card_list_nat @ ( sup_sup_set_list_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_list_nat @ A4 ) @ ( finite_card_list_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_3950_card__Un__le,axiom,
    ! [A4: set_set_nat,B5: set_set_nat] : ( ord_less_eq_nat @ ( finite_card_set_nat @ ( sup_sup_set_set_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_set_nat @ A4 ) @ ( finite_card_set_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_3951_card__Un__le,axiom,
    ! [A4: set_nat,B5: set_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( plus_plus_nat @ ( finite_card_nat @ A4 ) @ ( finite_card_nat @ B5 ) ) ) ).

% card_Un_le
thf(fact_3952_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_3953_nat__mult__div__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( divide_divide_nat @ M2 @ N ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3954_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_3955_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_3956_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_3957_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_3958_divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3959_divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3960_le__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3961_le__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3962_divide__left__mono,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C2 )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3963_divide__left__mono,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C2 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3964_neg__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3965_neg__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3966_neg__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3967_neg__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3968_pos__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3969_pos__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3970_pos__le__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C2 ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3971_pos__le__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C2 ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3972_mult__imp__div__pos__le,axiom,
    ! [Y: real,X: real,Z3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X @ ( times_times_real @ Z3 @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3973_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X @ ( times_times_rat @ Z3 @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ Z3 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3974_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z3: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z3 @ Y ) @ X )
       => ( ord_less_eq_real @ Z3 @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3975_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z3: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ Y ) @ X )
       => ( ord_less_eq_rat @ Z3 @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3976_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C2 @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C2 @ A ) @ ( divide_divide_real @ C2 @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3977_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C2 @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C2 @ A ) @ ( divide_divide_rat @ C2 @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3978_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_3979_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_3980_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_3981_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_3982_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3983_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3984_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3985_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3986_frac__le__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_3987_frac__le__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_3988_frac__less__eq,axiom,
    ! [Y: real,Z3: real,X: real,W2: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z3 != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X @ Y ) @ ( divide_divide_real @ W2 @ Z3 ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X @ Z3 ) @ ( times_times_real @ W2 @ Y ) ) @ ( times_times_real @ Y @ Z3 ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_3989_frac__less__eq,axiom,
    ! [Y: rat,Z3: rat,X: rat,W2: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z3 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X @ Y ) @ ( divide_divide_rat @ W2 @ Z3 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X @ Z3 ) @ ( times_times_rat @ W2 @ Y ) ) @ ( times_times_rat @ Y @ Z3 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_3990_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_3991_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_3992_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_3993_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_3994_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_3995_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_3996_four__x__squared,axiom,
    ! [X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_3997_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C2: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C2 ) ) @ ( 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 @ C2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_3998_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3999_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_4000_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_4001_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_4002_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_4003_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_4004_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_4005_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_4006_scaling__mono,axiom,
    ! [U: real,V2: real,R3: real,S2: real] :
      ( ( ord_less_eq_real @ U @ V2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ R3 )
       => ( ( ord_less_eq_real @ R3 @ S2 )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V2 @ U ) ) @ S2 ) ) @ V2 ) ) ) ) ).

% scaling_mono
thf(fact_4007_scaling__mono,axiom,
    ! [U: rat,V2: rat,R3: rat,S2: rat] :
      ( ( ord_less_eq_rat @ U @ V2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S2 )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V2 @ U ) ) @ S2 ) ) @ V2 ) ) ) ) ).

% scaling_mono
thf(fact_4008_nat__approx__posE,axiom,
    ! [E: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_4009_nat__approx__posE,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ~ ! [N3: nat] :
            ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E ) ) ).

% nat_approx_posE
thf(fact_4010_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_4011_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_4012_triangle__def,axiom,
    ( nat_triangle
    = ( ^ [N4: nat] : ( divide_divide_nat @ ( times_times_nat @ N4 @ ( suc @ N4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% triangle_def
thf(fact_4013_arith__geo__mean,axiom,
    ! [U: real,X: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_4014_arith__geo__mean,axiom,
    ! [U: rat,X: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_4015_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_4016_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_4017_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_4018_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_4019_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_4020_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_4021_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_4022_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_4023_insert_H__correct,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( vEBT_set_vebt @ ( vEBT_VEBT_insert @ T @ X ) )
        = ( inf_inf_set_nat @ ( sup_sup_set_nat @ ( vEBT_set_vebt @ T ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ).

% insert'_correct
thf(fact_4024_div__mult__self__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ M2 @ N ) @ N )
        = M2 ) ) ).

% div_mult_self_is_m
thf(fact_4025_div__mult__self1__is__m,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( divide_divide_nat @ ( times_times_nat @ N @ M2 ) @ N )
        = M2 ) ) ).

% div_mult_self1_is_m
thf(fact_4026_div__mult__self1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_4027_div__mult__self1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_4028_div__mult__self2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_4029_div__mult__self2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_4030_div__mult__self3,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_4031_div__mult__self3,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_4032_zdiv__numeral__Bit0,axiom,
    ! [V2: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_4033_real__divide__square__eq,axiom,
    ! [R3: real,A: real] :
      ( ( divide_divide_real @ ( times_times_real @ R3 @ A ) @ ( times_times_real @ R3 @ R3 ) )
      = ( divide_divide_real @ A @ R3 ) ) ).

% real_divide_square_eq
thf(fact_4034_div__mult__mult1__if,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ( C2 = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
          = zero_zero_nat ) )
      & ( ( C2 != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4035_div__mult__mult1__if,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ( C2 = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
          = zero_zero_int ) )
      & ( ( C2 != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_4036_div__mult__mult2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_4037_div__mult__mult2,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_4038_div__mult__mult1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_4039_div__mult__mult1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_4040_div__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( divide_divide_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = M2 ) ).

% div_by_Suc_0
thf(fact_4041_div__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_4042_div__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
        = ( plus_plus_nat @ C2 @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_4043_div__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
        = ( plus_plus_int @ C2 @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_4044_zdiv__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% zdiv_zmult2_eq
thf(fact_4045_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_4046_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_4047_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_4048_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_4049_atLeastAtMostPlus1__int__conv,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_eq_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
     => ( ( set_or1266510415728281911st_int @ M2 @ ( plus_plus_int @ one_one_int @ N ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N ) @ ( set_or1266510415728281911st_int @ M2 @ N ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_4050_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_4051_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_4052_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_4053_pos__imp__zdiv__pos__iff,axiom,
    ! [K2: int,I2: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K2 ) )
        = ( ord_less_eq_int @ K2 @ I2 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_4054_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_4055_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_4056_div__positive__int,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ L @ K2 )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) ) ) ) ).

% div_positive_int
thf(fact_4057_div__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K2 @ L ) )
      = ( ( K2 = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K2 )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K2 @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_4058_zdiv__mono2__neg,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B2 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_4059_zdiv__mono1__neg,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A2 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_4060_zdiv__eq__0__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( ( divide_divide_int @ I2 @ K2 )
        = zero_zero_int )
      = ( ( K2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K2 ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K2 @ I2 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_4061_zdiv__mono2,axiom,
    ! [A: int,B2: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B2 )
       => ( ( ord_less_eq_int @ B2 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B2 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_4062_zdiv__mono1,axiom,
    ! [A: int,A2: int,B: int] :
      ( ( ord_less_eq_int @ A @ A2 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A2 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_4063_int__div__less__self,axiom,
    ! [X: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ( ord_less_int @ one_one_int @ K2 )
       => ( ord_less_int @ ( divide_divide_int @ X @ K2 ) @ X ) ) ) ).

% int_div_less_self
thf(fact_4064_periodic__finite__ex,axiom,
    ! [D: int,P2: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X3: int,K: int] :
            ( ( P2 @ X3 )
            = ( P2 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D ) ) ) )
       => ( ( ? [X7: int] : ( P2 @ X7 ) )
          = ( ? [X4: int] :
                ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P2 @ X4 ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_4065_split__zdiv,axiom,
    ! [P2: int > $o,N: int,K2: int] :
      ( ( P2 @ ( divide_divide_int @ N @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P2 @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I ) ) ) ) ) ).

% split_zdiv
thf(fact_4066_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_neg_eq
thf(fact_4067_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q2 ) ) ) ) ).

% int_div_pos_eq
thf(fact_4068_cpmi,axiom,
    ! [D3: int,P2: int > $o,P6: int > $o,B5: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ X3 @ Z5 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ B5 )
                     => ( X3
                       != ( plus_plus_int @ Xb2 @ Xa ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( minus_minus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
           => ( ( ? [X7: int] : ( P2 @ X7 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ B5 )
                        & ( P2 @ ( plus_plus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_4069_cppi,axiom,
    ! [D3: int,P2: int > $o,P6: int > $o,A4: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D3 )
     => ( ? [Z5: int] :
          ! [X3: int] :
            ( ( ord_less_int @ Z5 @ X3 )
           => ( ( P2 @ X3 )
              = ( P6 @ X3 ) ) )
       => ( ! [X3: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                 => ! [Xb2: int] :
                      ( ( member_int @ Xb2 @ A4 )
                     => ( X3
                       != ( minus_minus_int @ Xb2 @ Xa ) ) ) )
             => ( ( P2 @ X3 )
               => ( P2 @ ( plus_plus_int @ X3 @ D3 ) ) ) )
         => ( ! [X3: int,K: int] :
                ( ( P6 @ X3 )
                = ( P6 @ ( minus_minus_int @ X3 @ ( times_times_int @ K @ D3 ) ) ) )
           => ( ( ? [X7: int] : ( P2 @ X7 ) )
              = ( ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ( P6 @ X4 ) )
                | ? [X4: int] :
                    ( ( member_int @ X4 @ ( set_or1266510415728281911st_int @ one_one_int @ D3 ) )
                    & ? [Y4: int] :
                        ( ( member_int @ Y4 @ A4 )
                        & ( P2 @ ( minus_minus_int @ Y4 @ X4 ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_4070_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_4071_div__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( divide_divide_int @ K2 @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K2 @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_4072_div__le__dividend,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ).

% div_le_dividend
thf(fact_4073_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_4074_div__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( divide_divide_nat @ M2 @ ( times_times_nat @ N @ Q2 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M2 @ N ) @ Q2 ) ) ).

% div_mult2_eq
thf(fact_4075_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_4076_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_4077_div__mult2__eq_H,axiom,
    ! [A: int,M2: nat,N: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% div_mult2_eq'
thf(fact_4078_div__mult2__eq_H,axiom,
    ! [A: nat,M2: nat,N: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).

% div_mult2_eq'
thf(fact_4079_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( divide_divide_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M2 @ N )
        | ( N = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_4080_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_4081_less__mult__imp__div__less,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( times_times_nat @ I2 @ N ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ I2 ) ) ).

% less_mult_imp_div_less
thf(fact_4082_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_4083_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_4084_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_4085_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_4086_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_4087_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_4088_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_4089_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_4090_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_4091_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_4092_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_4093_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_4094_div__less__iff__less__mult,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M2 @ Q2 ) @ N )
        = ( ord_less_nat @ M2 @ ( times_times_nat @ N @ Q2 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_4095_div__less__dividend,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( divide_divide_nat @ M2 @ N ) @ M2 ) ) ) ).

% div_less_dividend
thf(fact_4096_div__eq__dividend__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ( divide_divide_nat @ M2 @ N )
          = M2 )
        = ( N = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_4097_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N4: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M6 @ N4 )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N4 ) @ N4 ) ) ) ) ) ).

% div_if
thf(fact_4098_div__nat__eqI,axiom,
    ! [N: nat,Q2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q2 ) @ M2 )
     => ( ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q2 ) ) )
       => ( ( divide_divide_nat @ M2 @ N )
          = Q2 ) ) ) ).

% div_nat_eqI
thf(fact_4099_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q2 )
     => ( ( ord_less_eq_nat @ M2 @ ( divide_divide_nat @ N @ Q2 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M2 @ Q2 ) @ N ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_4100_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_4101_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_4102_split__div,axiom,
    ! [P2: nat > $o,M2: nat,N: nat] :
      ( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( ( ord_less_nat @ J @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J ) )
               => ( P2 @ I ) ) ) ) ) ) ).

% split_div
thf(fact_4103_power__diff__power__eq,axiom,
    ! [A: code_integer,N: nat,M2: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M2 @ N ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N @ M2 )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M2 ) @ ( power_8256067586552552935nteger @ A @ N ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4104_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_4105_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_4106_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_4107_split__div_H,axiom,
    ! [P2: nat > $o,M2: nat,N: nat] :
      ( ( P2 @ ( divide_divide_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
          & ( P2 @ zero_zero_nat ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N @ Q5 ) @ M2 )
            & ( ord_less_nat @ M2 @ ( times_times_nat @ N @ ( suc @ Q5 ) ) )
            & ( P2 @ Q5 ) ) ) ) ).

% split_div'
thf(fact_4108_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_4109_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_4110_linear__plus__1__le__power,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X @ one_one_real ) @ N ) ) ) ).

% linear_plus_1_le_power
thf(fact_4111_succ__list__to__short,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% succ_list_to_short
thf(fact_4112_pred__list__to__short,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = none_nat ) ) ) ) ).

% pred_list_to_short
thf(fact_4113_set__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_4114_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_4115_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_4116_case4_I11_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I4 ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X5: nat] :
              ( ( ( ( vEBT_VEBT_high @ X5 @ na )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X5 )
                & ( ord_less_eq_nat @ X5 @ ma ) ) ) ) ) ) ).

% case4(11)
thf(fact_4117_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_4118_le__sup__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_set_nat @ X @ Z3 )
        & ( ord_less_eq_set_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_4119_le__sup__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_rat @ X @ Z3 )
        & ( ord_less_eq_rat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_4120_le__sup__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_nat @ X @ Z3 )
        & ( ord_less_eq_nat @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_4121_le__sup__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z3 )
      = ( ( ord_less_eq_int @ X @ Z3 )
        & ( ord_less_eq_int @ Y @ Z3 ) ) ) ).

% le_sup_iff
thf(fact_4122_sup_Obounded__iff,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_set_nat @ B @ A )
        & ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_4123_sup_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_4124_sup_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_4125_sup_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.bounded_iff
thf(fact_4126_bit__split__inv,axiom,
    ! [X: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D ) @ ( vEBT_VEBT_low @ X @ D ) @ D )
      = X ) ).

% bit_split_inv
thf(fact_4127_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_4128_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X4: nat,N4: nat] : ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% high_def
thf(fact_4129_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_4130_high__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = Y ) ) ).

% high_inv
thf(fact_4131_low__inv,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) @ X ) @ N )
        = X ) ) ).

% low_inv
thf(fact_4132_le__inf__iff,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) )
      = ( ( ord_le3146513528884898305at_nat @ X @ Y )
        & ( ord_le3146513528884898305at_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_4133_le__inf__iff,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) )
      = ( ( ord_less_eq_set_nat @ X @ Y )
        & ( ord_less_eq_set_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_4134_le__inf__iff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) )
      = ( ( ord_less_eq_rat @ X @ Y )
        & ( ord_less_eq_rat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_4135_le__inf__iff,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) )
      = ( ( ord_less_eq_nat @ X @ Y )
        & ( ord_less_eq_nat @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_4136_le__inf__iff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z3 ) )
      = ( ( ord_less_eq_int @ X @ Y )
        & ( ord_less_eq_int @ X @ Z3 ) ) ) ).

% le_inf_iff
thf(fact_4137_inf_Obounded__iff,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
      = ( ( ord_le3146513528884898305at_nat @ A @ B )
        & ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_4138_inf_Obounded__iff,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
      = ( ( ord_less_eq_set_nat @ A @ B )
        & ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_4139_inf_Obounded__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_4140_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_4141_inf_Obounded__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.bounded_iff
thf(fact_4142_both__member__options__ding,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_ding
thf(fact_4143_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ Deg )
     => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X = Mi )
          | ( X = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_4144_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X = Mi )
          | ( X = Ma )
          | ( ( ord_less_nat @ X @ Ma )
            & ( ord_less_nat @ Mi @ X )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_4145_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_4146_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( 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 @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_4147_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ X @ ( 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 @ X @ N ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_4148_inf__sup__ord_I2_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4149_inf__sup__ord_I2_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4150_inf__sup__ord_I2_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4151_inf__sup__ord_I2_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4152_inf__sup__ord_I2_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_sup_ord(2)
thf(fact_4153_inf__sup__ord_I1_J,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_4154_inf__sup__ord_I1_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_4155_inf__sup__ord_I1_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_4156_inf__sup__ord_I1_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_4157_inf__sup__ord_I1_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_sup_ord(1)
thf(fact_4158_inf__le1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_4159_inf__le1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_4160_inf__le1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_4161_inf__le1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_4162_inf__le1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ X ) ).

% inf_le1
thf(fact_4163_inf__le2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_4164_inf__le2,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_4165_inf__le2,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_4166_inf__le2,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_4167_inf__le2,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ ( inf_inf_int @ X @ Y ) @ Y ) ).

% inf_le2
thf(fact_4168_le__infE,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ X @ A )
         => ~ ( ord_le3146513528884898305at_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_4169_le__infE,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_set_nat @ X @ A )
         => ~ ( ord_less_eq_set_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_4170_le__infE,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) )
     => ~ ( ( ord_less_eq_rat @ X @ A )
         => ~ ( ord_less_eq_rat @ X @ B ) ) ) ).

% le_infE
thf(fact_4171_le__infE,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X @ A )
         => ~ ( ord_less_eq_nat @ X @ B ) ) ) ).

% le_infE
thf(fact_4172_le__infE,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) )
     => ~ ( ( ord_less_eq_int @ X @ A )
         => ~ ( ord_less_eq_int @ X @ B ) ) ) ).

% le_infE
thf(fact_4173_le__infI,axiom,
    ! [X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ A )
     => ( ( ord_le3146513528884898305at_nat @ X @ B )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4174_le__infI,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ( ord_less_eq_set_nat @ X @ B )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4175_le__infI,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ( ord_less_eq_rat @ X @ B )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4176_le__infI,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ( ord_less_eq_nat @ X @ B )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4177_le__infI,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ( ord_less_eq_int @ X @ B )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ A @ B ) ) ) ) ).

% le_infI
thf(fact_4178_inf__mono,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,D: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ( ord_le3146513528884898305at_nat @ B @ D )
       => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ ( inf_in2572325071724192079at_nat @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_4179_inf__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_4180_inf__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ ( inf_inf_rat @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_4181_inf__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_4182_inf__mono,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ ( inf_inf_int @ C2 @ D ) ) ) ) ).

% inf_mono
thf(fact_4183_le__infI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_4184_le__infI1,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_4185_le__infI1,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_4186_le__infI1,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_4187_le__infI1,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI1
thf(fact_4188_le__infI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ X )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_4189_le__infI2,axiom,
    ! [B: set_nat,X: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ X )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_4190_le__infI2,axiom,
    ! [B: rat,X: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ X )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_4191_le__infI2,axiom,
    ! [B: nat,X: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_4192_le__infI2,axiom,
    ! [B: int,X: int,A: int] :
      ( ( ord_less_eq_int @ B @ X )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ X ) ) ).

% le_infI2
thf(fact_4193_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_4194_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_4195_inf_OorderE,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( A
        = ( inf_inf_rat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4196_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4197_inf_OorderE,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( A
        = ( inf_inf_int @ A @ B ) ) ) ).

% inf.orderE
thf(fact_4198_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_4199_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_4200_inf_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( inf_inf_rat @ A @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% inf.orderI
thf(fact_4201_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_4202_inf_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( inf_inf_int @ A @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% inf.orderI
thf(fact_4203_inf__unique,axiom,
    ! [F2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat,Z: set_Pr1261947904930325089at_nat] :
              ( ( ord_le3146513528884898305at_nat @ X3 @ Y3 )
             => ( ( ord_le3146513528884898305at_nat @ X3 @ Z )
               => ( ord_le3146513528884898305at_nat @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_in2572325071724192079at_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4204_inf__unique,axiom,
    ! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
              ( ( ord_less_eq_set_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_set_nat @ X3 @ Z )
               => ( ord_less_eq_set_nat @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_inf_set_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4205_inf__unique,axiom,
    ! [F2: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: rat,Y3: rat,Z: rat] :
              ( ( ord_less_eq_rat @ X3 @ Y3 )
             => ( ( ord_less_eq_rat @ X3 @ Z )
               => ( ord_less_eq_rat @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_inf_rat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4206_inf__unique,axiom,
    ! [F2: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( ( ord_less_eq_nat @ X3 @ Y3 )
             => ( ( ord_less_eq_nat @ X3 @ Z )
               => ( ord_less_eq_nat @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_inf_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4207_inf__unique,axiom,
    ! [F2: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y3 ) @ X3 )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ ( F2 @ X3 @ Y3 ) @ Y3 )
       => ( ! [X3: int,Y3: int,Z: int] :
              ( ( ord_less_eq_int @ X3 @ Y3 )
             => ( ( ord_less_eq_int @ X3 @ Z )
               => ( ord_less_eq_int @ X3 @ ( F2 @ Y3 @ Z ) ) ) )
         => ( ( inf_inf_int @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% inf_unique
thf(fact_4208_le__iff__inf,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [X4: set_Pr1261947904930325089at_nat,Y4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4209_le__iff__inf,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( inf_inf_set_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4210_le__iff__inf,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( inf_inf_rat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4211_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( inf_inf_nat @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4212_le__iff__inf,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( inf_inf_int @ X4 @ Y4 )
          = X4 ) ) ) ).

% le_iff_inf
thf(fact_4213_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_4214_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_4215_inf_Oabsorb1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( inf_inf_rat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4216_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4217_inf_Oabsorb1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( inf_inf_int @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_4218_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_4219_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_4220_inf_Oabsorb2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( inf_inf_rat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4221_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4222_inf_Oabsorb2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( inf_inf_int @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_4223_inf__absorb1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_4224_inf__absorb1,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( inf_inf_set_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_4225_inf__absorb1,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( inf_inf_rat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_4226_inf__absorb1,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( inf_inf_nat @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_4227_inf__absorb1,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( inf_inf_int @ X @ Y )
        = X ) ) ).

% inf_absorb1
thf(fact_4228_inf__absorb2,axiom,
    ! [Y: set_Pr1261947904930325089at_nat,X: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ Y @ X )
     => ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4229_inf__absorb2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( inf_inf_set_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4230_inf__absorb2,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( inf_inf_rat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4231_inf__absorb2,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( inf_inf_nat @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4232_inf__absorb2,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( inf_inf_int @ X @ Y )
        = Y ) ) ).

% inf_absorb2
thf(fact_4233_inf_OboundedE,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) )
     => ~ ( ( ord_le3146513528884898305at_nat @ A @ B )
         => ~ ( ord_le3146513528884898305at_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_4234_inf_OboundedE,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_set_nat @ A @ B )
         => ~ ( ord_less_eq_set_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_4235_inf_OboundedE,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_rat @ A @ B )
         => ~ ( ord_less_eq_rat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_4236_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_4237_inf_OboundedE,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) )
     => ~ ( ( ord_less_eq_int @ A @ B )
         => ~ ( ord_less_eq_int @ A @ C2 ) ) ) ).

% inf.boundedE
thf(fact_4238_inf_OboundedI,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ B )
     => ( ( ord_le3146513528884898305at_nat @ A @ C2 )
       => ( ord_le3146513528884898305at_nat @ A @ ( inf_in2572325071724192079at_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_4239_inf_OboundedI,axiom,
    ! [A: set_nat,B: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ B )
     => ( ( ord_less_eq_set_nat @ A @ C2 )
       => ( ord_less_eq_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_4240_inf_OboundedI,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ A @ C2 )
       => ( ord_less_eq_rat @ A @ ( inf_inf_rat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_4241_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C2 )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_4242_inf_OboundedI,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ A @ C2 )
       => ( ord_less_eq_int @ A @ ( inf_inf_int @ B @ C2 ) ) ) ) ).

% inf.boundedI
thf(fact_4243_inf__greatest,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X @ Y )
     => ( ( ord_le3146513528884898305at_nat @ X @ Z3 )
       => ( ord_le3146513528884898305at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_4244_inf__greatest,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_less_eq_set_nat @ X @ Z3 )
       => ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_4245_inf__greatest,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_less_eq_rat @ X @ Z3 )
       => ( ord_less_eq_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_4246_inf__greatest,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Z3 )
       => ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_4247_inf__greatest,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_less_eq_int @ X @ Z3 )
       => ( ord_less_eq_int @ X @ ( inf_inf_int @ Y @ Z3 ) ) ) ) ).

% inf_greatest
thf(fact_4248_inf_Oorder__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( A5
          = ( inf_in2572325071724192079at_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_4249_inf_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( A5
          = ( inf_inf_set_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_4250_inf_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( A5
          = ( inf_inf_rat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_4251_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( A5
          = ( inf_inf_nat @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_4252_inf_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( A5
          = ( inf_inf_int @ A5 @ B4 ) ) ) ) ).

% inf.order_iff
thf(fact_4253_inf_Ocobounded1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4254_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_4255_inf_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4256_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4257_inf_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_4258_inf_Ocobounded2,axiom,
    ! [A: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4259_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_4260_inf_Ocobounded2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4261_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4262_inf_Ocobounded2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_4263_inf_Oabsorb__iff1,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B4: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_4264_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_4265_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( inf_inf_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_4266_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( inf_inf_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_4267_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( inf_inf_int @ A5 @ B4 )
          = A5 ) ) ) ).

% inf.absorb_iff1
thf(fact_4268_inf_Oabsorb__iff2,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [B4: set_Pr1261947904930325089at_nat,A5: set_Pr1261947904930325089at_nat] :
          ( ( inf_in2572325071724192079at_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_4269_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( inf_inf_set_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_4270_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( inf_inf_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_4271_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( inf_inf_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_4272_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( inf_inf_int @ A5 @ B4 )
          = B4 ) ) ) ).

% inf.absorb_iff2
thf(fact_4273_inf_OcoboundedI1,axiom,
    ! [A: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,B: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_4274_inf_OcoboundedI1,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_4275_inf_OcoboundedI1,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_4276_inf_OcoboundedI1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_4277_inf_OcoboundedI1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI1
thf(fact_4278_inf_OcoboundedI2,axiom,
    ! [B: set_Pr1261947904930325089at_nat,C2: set_Pr1261947904930325089at_nat,A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ B @ C2 )
     => ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_4279_inf_OcoboundedI2,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ C2 )
     => ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_4280_inf_OcoboundedI2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ C2 )
     => ( ord_less_eq_rat @ ( inf_inf_rat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_4281_inf_OcoboundedI2,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C2 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_4282_inf_OcoboundedI2,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ B @ C2 )
     => ( ord_less_eq_int @ ( inf_inf_int @ A @ B ) @ C2 ) ) ).

% inf.coboundedI2
thf(fact_4283_sup_OcoboundedI2,axiom,
    ! [C2: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ B )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_4284_sup_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ B )
     => ( ord_less_eq_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_4285_sup_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_4286_sup_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI2
thf(fact_4287_sup_OcoboundedI1,axiom,
    ! [C2: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_4288_sup_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ord_less_eq_rat @ C2 @ ( sup_sup_rat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_4289_sup_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_4290_sup_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).

% sup.coboundedI1
thf(fact_4291_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_4292_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( sup_sup_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_4293_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_4294_sup_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( sup_sup_int @ A5 @ B4 )
          = B4 ) ) ) ).

% sup.absorb_iff2
thf(fact_4295_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( ( sup_sup_set_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_4296_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( sup_sup_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_4297_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( sup_sup_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_4298_sup_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( sup_sup_int @ A5 @ B4 )
          = A5 ) ) ) ).

% sup.absorb_iff1
thf(fact_4299_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_4300_sup_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_4301_sup_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded2
thf(fact_4302_sup_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded2
thf(fact_4303_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_4304_sup_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( sup_sup_rat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_4305_sup_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( sup_sup_nat @ A @ B ) ) ).

% sup.cobounded1
thf(fact_4306_sup_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( sup_sup_int @ A @ B ) ) ).

% sup.cobounded1
thf(fact_4307_sup_Oorder__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [B4: set_nat,A5: set_nat] :
          ( A5
          = ( sup_sup_set_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_4308_sup_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( sup_sup_rat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_4309_sup_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( sup_sup_nat @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_4310_sup_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( sup_sup_int @ A5 @ B4 ) ) ) ) ).

% sup.order_iff
thf(fact_4311_sup_OboundedI,axiom,
    ! [B: set_nat,A: set_nat,C2: set_nat] :
      ( ( ord_less_eq_set_nat @ B @ A )
     => ( ( ord_less_eq_set_nat @ C2 @ A )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_4312_sup_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ A )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_4313_sup_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_4314_sup_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A ) ) ) ).

% sup.boundedI
thf(fact_4315_sup_OboundedE,axiom,
    ! [B: set_nat,C2: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_set_nat @ B @ A )
         => ~ ( ord_less_eq_set_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_4316_sup_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_4317_sup_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_4318_sup_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% sup.boundedE
thf(fact_4319_sup__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( sup_sup_set_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_4320_sup__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( sup_sup_rat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_4321_sup__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( sup_sup_nat @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_4322_sup__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( sup_sup_int @ X @ Y )
        = Y ) ) ).

% sup_absorb2
thf(fact_4323_sup__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( sup_sup_set_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_4324_sup__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( sup_sup_rat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_4325_sup__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( sup_sup_nat @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_4326_sup__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( sup_sup_int @ X @ Y )
        = X ) ) ).

% sup_absorb1
thf(fact_4327_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_4328_sup_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( sup_sup_rat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_4329_sup_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_4330_sup_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( sup_sup_int @ A @ B )
        = B ) ) ).

% sup.absorb2
thf(fact_4331_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_4332_sup_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( sup_sup_rat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_4333_sup_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_4334_sup_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( sup_sup_int @ A @ B )
        = A ) ) ).

% sup.absorb1
thf(fact_4335_sup__unique,axiom,
    ! [F2: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
      ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: set_nat,Y3: set_nat] : ( ord_less_eq_set_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: set_nat,Y3: set_nat,Z: set_nat] :
              ( ( ord_less_eq_set_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_set_nat @ Z @ X3 )
               => ( ord_less_eq_set_nat @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
         => ( ( sup_sup_set_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_4336_sup__unique,axiom,
    ! [F2: rat > rat > rat,X: rat,Y: rat] :
      ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: rat,Y3: rat,Z: rat] :
              ( ( ord_less_eq_rat @ Y3 @ X3 )
             => ( ( ord_less_eq_rat @ Z @ X3 )
               => ( ord_less_eq_rat @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
         => ( ( sup_sup_rat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_4337_sup__unique,axiom,
    ! [F2: nat > nat > nat,X: nat,Y: nat] :
      ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: nat,Y3: nat,Z: nat] :
              ( ( ord_less_eq_nat @ Y3 @ X3 )
             => ( ( ord_less_eq_nat @ Z @ X3 )
               => ( ord_less_eq_nat @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
         => ( ( sup_sup_nat @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_4338_sup__unique,axiom,
    ! [F2: int > int > int,X: int,Y: int] :
      ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
     => ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
       => ( ! [X3: int,Y3: int,Z: int] :
              ( ( ord_less_eq_int @ Y3 @ X3 )
             => ( ( ord_less_eq_int @ Z @ X3 )
               => ( ord_less_eq_int @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
         => ( ( sup_sup_int @ X @ Y )
            = ( F2 @ X @ Y ) ) ) ) ) ).

% sup_unique
thf(fact_4339_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_4340_sup_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( sup_sup_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% sup.orderI
thf(fact_4341_sup_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( sup_sup_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% sup.orderI
thf(fact_4342_sup_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( sup_sup_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% sup.orderI
thf(fact_4343_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_4344_sup_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( sup_sup_rat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_4345_sup_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( sup_sup_nat @ A @ B ) ) ) ).

% sup.orderE
thf(fact_4346_sup_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( sup_sup_int @ A @ B ) ) ) ).

% sup.orderE
thf(fact_4347_le__iff__sup,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [X4: set_nat,Y4: set_nat] :
          ( ( sup_sup_set_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_4348_le__iff__sup,axiom,
    ( ord_less_eq_rat
    = ( ^ [X4: rat,Y4: rat] :
          ( ( sup_sup_rat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_4349_le__iff__sup,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y4: nat] :
          ( ( sup_sup_nat @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_4350_le__iff__sup,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Y4: int] :
          ( ( sup_sup_int @ X4 @ Y4 )
          = Y4 ) ) ) ).

% le_iff_sup
thf(fact_4351_sup__least,axiom,
    ! [Y: set_nat,X: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_less_eq_set_nat @ Z3 @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_4352_sup__least,axiom,
    ! [Y: rat,X: rat,Z3: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_less_eq_rat @ Z3 @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_4353_sup__least,axiom,
    ! [Y: nat,X: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ Z3 @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_4354_sup__least,axiom,
    ! [Y: int,X: int,Z3: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_less_eq_int @ Z3 @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z3 ) @ X ) ) ) ).

% sup_least
thf(fact_4355_sup__mono,axiom,
    ! [A: set_nat,C2: set_nat,B: set_nat,D: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ C2 )
     => ( ( ord_less_eq_set_nat @ B @ D )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_4356_sup__mono,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ C2 )
     => ( ( ord_less_eq_rat @ B @ D )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ ( sup_sup_rat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_4357_sup__mono,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ C2 )
     => ( ( ord_less_eq_nat @ B @ D )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ ( sup_sup_nat @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_4358_sup__mono,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( ord_less_eq_int @ A @ C2 )
     => ( ( ord_less_eq_int @ B @ D )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ ( sup_sup_int @ C2 @ D ) ) ) ) ).

% sup_mono
thf(fact_4359_sup_Omono,axiom,
    ! [C2: set_nat,A: set_nat,D: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ C2 @ A )
     => ( ( ord_less_eq_set_nat @ D @ B )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C2 @ D ) @ ( sup_sup_set_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_4360_sup_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ C2 @ D ) @ ( sup_sup_rat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_4361_sup_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ C2 @ D ) @ ( sup_sup_nat @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_4362_sup_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( sup_sup_int @ C2 @ D ) @ ( sup_sup_int @ A @ B ) ) ) ) ).

% sup.mono
thf(fact_4363_le__supI2,axiom,
    ! [X: set_nat,B: set_nat,A: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ B )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_4364_le__supI2,axiom,
    ! [X: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ X @ B )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI2
thf(fact_4365_le__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ X @ B )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI2
thf(fact_4366_le__supI2,axiom,
    ! [X: int,B: int,A: int] :
      ( ( ord_less_eq_int @ X @ B )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI2
thf(fact_4367_le__supI1,axiom,
    ! [X: set_nat,A: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ A )
     => ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_4368_le__supI1,axiom,
    ! [X: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ X @ A )
     => ( ord_less_eq_rat @ X @ ( sup_sup_rat @ A @ B ) ) ) ).

% le_supI1
thf(fact_4369_le__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X @ A )
     => ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% le_supI1
thf(fact_4370_le__supI1,axiom,
    ! [X: int,A: int,B: int] :
      ( ( ord_less_eq_int @ X @ A )
     => ( ord_less_eq_int @ X @ ( sup_sup_int @ A @ B ) ) ) ).

% le_supI1
thf(fact_4371_sup__ge2,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_4372_sup__ge2,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge2
thf(fact_4373_sup__ge2,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge2
thf(fact_4374_sup__ge2,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge2
thf(fact_4375_sup__ge1,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_4376_sup__ge1,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% sup_ge1
thf(fact_4377_sup__ge1,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% sup_ge1
thf(fact_4378_sup__ge1,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% sup_ge1
thf(fact_4379_le__supI,axiom,
    ! [A: set_nat,X: set_nat,B: set_nat] :
      ( ( ord_less_eq_set_nat @ A @ X )
     => ( ( ord_less_eq_set_nat @ B @ X )
       => ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_4380_le__supI,axiom,
    ! [A: rat,X: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ X )
     => ( ( ord_less_eq_rat @ B @ X )
       => ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_4381_le__supI,axiom,
    ! [A: nat,X: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X )
     => ( ( ord_less_eq_nat @ B @ X )
       => ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_4382_le__supI,axiom,
    ! [A: int,X: int,B: int] :
      ( ( ord_less_eq_int @ A @ X )
     => ( ( ord_less_eq_int @ B @ X )
       => ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X ) ) ) ).

% le_supI
thf(fact_4383_le__supE,axiom,
    ! [A: set_nat,B: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_set_nat @ A @ X )
         => ~ ( ord_less_eq_set_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_4384_le__supE,axiom,
    ! [A: rat,B: rat,X: rat] :
      ( ( ord_less_eq_rat @ ( sup_sup_rat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_rat @ A @ X )
         => ~ ( ord_less_eq_rat @ B @ X ) ) ) ).

% le_supE
thf(fact_4385_le__supE,axiom,
    ! [A: nat,B: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( sup_sup_nat @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_nat @ A @ X )
         => ~ ( ord_less_eq_nat @ B @ X ) ) ) ).

% le_supE
thf(fact_4386_le__supE,axiom,
    ! [A: int,B: int,X: int] :
      ( ( ord_less_eq_int @ ( sup_sup_int @ A @ B ) @ X )
     => ~ ( ( ord_less_eq_int @ A @ X )
         => ~ ( ord_less_eq_int @ B @ X ) ) ) ).

% le_supE
thf(fact_4387_inf__sup__ord_I3_J,axiom,
    ! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_4388_inf__sup__ord_I3_J,axiom,
    ! [X: rat,Y: rat] : ( ord_less_eq_rat @ X @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_4389_inf__sup__ord_I3_J,axiom,
    ! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_4390_inf__sup__ord_I3_J,axiom,
    ! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(3)
thf(fact_4391_inf__sup__ord_I4_J,axiom,
    ! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_4392_inf__sup__ord_I4_J,axiom,
    ! [Y: rat,X: rat] : ( ord_less_eq_rat @ Y @ ( sup_sup_rat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_4393_inf__sup__ord_I4_J,axiom,
    ! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_4394_inf__sup__ord_I4_J,axiom,
    ! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).

% inf_sup_ord(4)
thf(fact_4395_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ 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 @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_4396_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X3: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X3 @ N ) )
     => ( ( vEBT_invar_vebt @ Summary @ M2 )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( 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 ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ 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 @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
                                & ! [X3: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X3 @ N )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
                                   => ( ( ord_less_nat @ Mi @ X3 )
                                      & ( ord_less_eq_nat @ X3 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_4397_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A3: $o,B3: $o] :
              ( A12
              = ( vEBT_Leaf @ A3 @ B3 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ( vEBT_invar_vebt @ X5 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary2 @ M )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                     => ( ( M = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                           => ~ ! [X5: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ( vEBT_invar_vebt @ X5 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary2 @ M )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                       => ( ( M
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
                             => ~ ! [X5: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_invar_vebt @ X5 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary2 @ M )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                         => ( ( M = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X5: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X5: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi2 @ X5 )
                                                      & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X5: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ( vEBT_invar_vebt @ X5 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary2 @ M )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                           => ( ( M
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                     => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ X7 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X5: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X5: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X5 @ N3 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X5 )
                                                        & ( ord_less_eq_nat @ X5 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_4398_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A13: vEBT_VEBT,A24: nat] :
          ( ( ? [A5: $o,B4: $o] :
                ( A13
                = ( vEBT_Leaf @ A5 @ B4 ) )
            & ( A24
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X7 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT] :
              ( ( A13
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A24 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary4 @ X7 )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary4 @ N4 )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ N4 ) )
              & ! [I: nat] :
                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                              = I )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList4: list_VEBT_VEBT,N4: nat,Summary4: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A13
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A24 @ TreeList4 @ Summary4 ) )
              & ! [X4: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                 => ( vEBT_invar_vebt @ X4 @ N4 ) )
              & ( vEBT_invar_vebt @ Summary4 @ ( suc @ N4 ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList4 )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
              & ( A24
                = ( plus_plus_nat @ N4 @ ( suc @ N4 ) ) )
              & ! [I: nat] :
                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ X7 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary4 @ I ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X4: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList4 ) )
                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A24 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N4 )
                          = I )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N4 ) ) )
                      & ! [X4: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X4 @ N4 )
                              = I )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ I ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_4399_distrib__inf__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ ( inf_in2572325071724192079at_nat @ X @ Z3 ) ) @ ( inf_in2572325071724192079at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_4400_distrib__inf__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z3 ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_4401_distrib__inf__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ ( inf_inf_rat @ X @ Y ) @ ( inf_inf_rat @ X @ Z3 ) ) @ ( inf_inf_rat @ X @ ( sup_sup_rat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_4402_distrib__inf__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z3 ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_4403_distrib__inf__le,axiom,
    ! [X: int,Y: int,Z3: int] : ( ord_less_eq_int @ ( sup_sup_int @ ( inf_inf_int @ X @ Y ) @ ( inf_inf_int @ X @ Z3 ) ) @ ( inf_inf_int @ X @ ( sup_sup_int @ Y @ Z3 ) ) ) ).

% distrib_inf_le
thf(fact_4404_distrib__sup__le,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ ( sup_su6327502436637775413at_nat @ X @ ( inf_in2572325071724192079at_nat @ Y @ Z3 ) ) @ ( inf_in2572325071724192079at_nat @ ( sup_su6327502436637775413at_nat @ X @ Y ) @ ( sup_su6327502436637775413at_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_4405_distrib__sup__le,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z3 ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_4406_distrib__sup__le,axiom,
    ! [X: rat,Y: rat,Z3: rat] : ( ord_less_eq_rat @ ( sup_sup_rat @ X @ ( inf_inf_rat @ Y @ Z3 ) ) @ ( inf_inf_rat @ ( sup_sup_rat @ X @ Y ) @ ( sup_sup_rat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_4407_distrib__sup__le,axiom,
    ! [X: nat,Y: nat,Z3: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z3 ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_4408_distrib__sup__le,axiom,
    ! [X: int,Y: int,Z3: int] : ( ord_less_eq_int @ ( sup_sup_int @ X @ ( inf_inf_int @ Y @ Z3 ) ) @ ( inf_inf_int @ ( sup_sup_int @ X @ Y ) @ ( sup_sup_int @ X @ Z3 ) ) ) ).

% distrib_sup_le
thf(fact_4409_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N4: nat,TreeList4: list_VEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList4 @ ( vEBT_VEBT_high @ X4 @ N4 ) ) @ ( vEBT_VEBT_low @ X4 @ N4 ) ) ) ) ).

% in_children_def
thf(fact_4410_nested__mint,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat,Va3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( N
          = ( suc @ ( suc @ Va3 ) ) )
       => ( ~ ( 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 @ Va3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( suc @ ( divide_divide_nat @ Va3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ).

% nested_mint
thf(fact_4411_summaxma,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
     => ( ( Mi != Ma )
       => ( ( the_nat @ ( vEBT_vebt_maxt @ Summary ) )
          = ( vEBT_VEBT_high @ Ma @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% summaxma
thf(fact_4412_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_4413_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_4414_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_4415_del__x__mi__lets__in__not__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = 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 @ TreeList2 @ ( 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 @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ Newnode ) )
                     => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( 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_4416_del__x__not__mi__newnode__not__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
               => ( ~ ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Newlist
                      = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ Newnode ) )
                   => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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_4417_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_4418_length__list__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) )
      = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ).

% length_list_update
thf(fact_4419_length__list__update,axiom,
    ! [Xs2: list_o,I2: nat,X: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs2 @ I2 @ X ) )
      = ( size_size_list_o @ Xs2 ) ) ).

% length_list_update
thf(fact_4420_length__list__update,axiom,
    ! [Xs2: list_nat,I2: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_list_update
thf(fact_4421_length__list__update,axiom,
    ! [Xs2: list_int,I2: nat,X: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs2 @ I2 @ X ) )
      = ( size_size_list_int @ Xs2 ) ) ).

% length_list_update
thf(fact_4422_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4423_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4424_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_4425_list__update__beyond,axiom,
    ! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ I2 )
     => ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4426_list__update__beyond,axiom,
    ! [Xs2: list_o,I2: nat,X: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ I2 )
     => ( ( list_update_o @ Xs2 @ I2 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4427_list__update__beyond,axiom,
    ! [Xs2: list_nat,I2: nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ I2 )
     => ( ( list_update_nat @ Xs2 @ I2 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4428_list__update__beyond,axiom,
    ! [Xs2: list_int,I2: nat,X: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ I2 )
     => ( ( list_update_int @ Xs2 @ I2 @ X )
        = Xs2 ) ) ).

% list_update_beyond
thf(fact_4429_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_4430_nth__list__update__eq,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4431_nth__list__update__eq,axiom,
    ! [I2: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4432_nth__list__update__eq,axiom,
    ! [I2: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4433_nth__list__update__eq,axiom,
    ! [I2: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_4434_set__swap,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) @ J2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
          = ( set_VEBT_VEBT2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4435_set__swap,axiom,
    ! [I2: nat,Xs2: list_o,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J2 ) ) @ J2 @ ( nth_o @ Xs2 @ I2 ) ) )
          = ( set_o2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4436_set__swap,axiom,
    ! [I2: nat,Xs2: list_nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I2 ) ) )
          = ( set_nat2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4437_set__swap,axiom,
    ! [I2: nat,Xs2: list_int,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J2 ) ) @ J2 @ ( nth_int @ Xs2 @ I2 ) ) )
          = ( set_int2 @ Xs2 ) ) ) ) ).

% set_swap
thf(fact_4438_unique__quotient,axiom,
    ! [A: int,B: int,Q2: int,R3: int,Q4: int,R5: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R5 ) )
       => ( Q2 = Q4 ) ) ) ).

% unique_quotient
thf(fact_4439_unique__remainder,axiom,
    ! [A: int,B: int,Q2: int,R3: int,Q4: int,R5: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R5 ) )
       => ( R3 = R5 ) ) ) ).

% unique_remainder
thf(fact_4440_zip__update,axiom,
    ! [Xs2: list_VEBT_VEBT,I2: nat,X: vEBT_VEBT,Ys: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( zip_VE537291747668921783T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ ( list_u1324408373059187874T_VEBT @ Ys @ I2 @ Y ) )
      = ( list_u6961636818849549845T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs2 @ Ys ) @ I2 @ ( produc537772716801021591T_VEBT @ X @ Y ) ) ) ).

% zip_update
thf(fact_4441_zip__update,axiom,
    ! [Xs2: list_Code_integer,I2: nat,X: code_integer,Ys: list_Code_integer,Y: code_integer] :
      ( ( zip_Co3543743374963494515nteger @ ( list_u5447711078246177391nteger @ Xs2 @ I2 @ X ) @ ( list_u5447711078246177391nteger @ Ys @ I2 @ Y ) )
      = ( list_u2254550707601501961nteger @ ( zip_Co3543743374963494515nteger @ Xs2 @ Ys ) @ I2 @ ( produc1086072967326762835nteger @ X @ Y ) ) ) ).

% zip_update
thf(fact_4442_zip__update,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,I2: nat,X: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( zip_Pr4664179122662387191at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ I2 @ X ) @ ( list_u6180841689913720943at_nat @ Ys @ I2 @ Y ) )
      = ( list_u5003261594476800725at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs2 @ Ys ) @ I2 @ ( produc6161850002892822231at_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_4443_zip__update,axiom,
    ! [Xs2: list_s1210847774152347623at_nat,I2: nat,X: set_Pr1261947904930325089at_nat,Ys: list_s1210847774152347623at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( zip_se5600341670672612855at_nat @ ( list_u8444657558853818831at_nat @ Xs2 @ I2 @ X ) @ ( list_u8444657558853818831at_nat @ Ys @ I2 @ Y ) )
      = ( list_u4696772448584712917at_nat @ ( zip_se5600341670672612855at_nat @ Xs2 @ Ys ) @ I2 @ ( produc2922128104949294807at_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_4444_zip__update,axiom,
    ! [Xs2: list_nat,I2: nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( zip_nat_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ ( list_update_nat @ Ys @ I2 @ Y ) )
      = ( list_u6180841689913720943at_nat @ ( zip_nat_nat @ Xs2 @ Ys ) @ I2 @ ( product_Pair_nat_nat @ X @ Y ) ) ) ).

% zip_update
thf(fact_4445_zip__update,axiom,
    ! [Xs2: list_int,I2: nat,X: int,Ys: list_int,Y: int] :
      ( ( zip_int_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ ( list_update_int @ Ys @ I2 @ Y ) )
      = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs2 @ Ys ) @ I2 @ ( product_Pair_int_int @ X @ Y ) ) ) ).

% zip_update
thf(fact_4446_option_Osel,axiom,
    ! [X2: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_4447_option_Osel,axiom,
    ! [X2: nat] :
      ( ( the_nat @ ( some_nat @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_4448_option_Osel,axiom,
    ! [X2: num] :
      ( ( the_num @ ( some_num @ X2 ) )
      = X2 ) ).

% option.sel
thf(fact_4449_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_4450_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_4451_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_4452_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_4453_div__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( divide_divide_int @ K2 @ L )
        = Q2 ) ) ).

% div_int_unique
thf(fact_4454_list__update__code_I3_J,axiom,
    ! [X: nat,Xs2: list_nat,I2: nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ ( suc @ I2 ) @ Y )
      = ( cons_nat @ X @ ( list_update_nat @ Xs2 @ I2 @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4455_list__update__code_I3_J,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,I2: nat,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ ( suc @ I2 ) @ Y )
      = ( cons_VEBT_VEBT @ X @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ Y ) ) ) ).

% list_update_code(3)
thf(fact_4456_list__update__code_I2_J,axiom,
    ! [X: nat,Xs2: list_nat,Y: nat] :
      ( ( list_update_nat @ ( cons_nat @ X @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_nat @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4457_list__update__code_I2_J,axiom,
    ! [X: vEBT_VEBT,Xs2: list_VEBT_VEBT,Y: vEBT_VEBT] :
      ( ( list_u1324408373059187874T_VEBT @ ( cons_VEBT_VEBT @ X @ Xs2 ) @ zero_zero_nat @ Y )
      = ( cons_VEBT_VEBT @ Y @ Xs2 ) ) ).

% list_update_code(2)
thf(fact_4458_set__encode__eq,axiom,
    ! [A4: set_nat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( nat_set_encode @ A4 )
            = ( nat_set_encode @ B5 ) )
          = ( A4 = B5 ) ) ) ) ).

% set_encode_eq
thf(fact_4459_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4460_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4461_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_4462_eucl__rel__int__dividesI,axiom,
    ! [L: int,K2: int,Q2: int] :
      ( ( L != zero_zero_int )
     => ( ( K2
          = ( times_times_int @ Q2 @ L ) )
       => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_4463_set__update__memI,axiom,
    ! [N: nat,Xs2: list_complex,X: complex] :
      ( ( ord_less_nat @ N @ ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( member_complex @ X @ ( set_complex2 @ ( list_update_complex @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4464_set__update__memI,axiom,
    ! [N: nat,Xs2: list_real,X: real] :
      ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
     => ( member_real @ X @ ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4465_set__update__memI,axiom,
    ! [N: nat,Xs2: list_set_nat,X: set_nat] :
      ( ( ord_less_nat @ N @ ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( member_set_nat @ X @ ( set_set_nat2 @ ( list_update_set_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4466_set__update__memI,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4467_set__update__memI,axiom,
    ! [N: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( member_o @ X @ ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4468_set__update__memI,axiom,
    ! [N: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4469_set__update__memI,axiom,
    ! [N: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( member_int @ X @ ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_4470_list__update__same__conv,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X )
          = Xs2 )
        = ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4471_list__update__same__conv,axiom,
    ! [I2: nat,Xs2: list_o,X: $o] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ( list_update_o @ Xs2 @ I2 @ X )
          = Xs2 )
        = ( ( nth_o @ Xs2 @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4472_list__update__same__conv,axiom,
    ! [I2: nat,Xs2: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( list_update_nat @ Xs2 @ I2 @ X )
          = Xs2 )
        = ( ( nth_nat @ Xs2 @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4473_list__update__same__conv,axiom,
    ! [I2: nat,Xs2: list_int,X: int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( list_update_int @ Xs2 @ I2 @ X )
          = Xs2 )
        = ( ( nth_int @ Xs2 @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_4474_nth__list__update,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,J2: nat,X: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ( I2 = J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J2 )
            = X ) )
        & ( ( I2 != J2 )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ X ) @ J2 )
            = ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4475_nth__list__update,axiom,
    ! [I2: nat,Xs2: list_o,X: $o,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( list_update_o @ Xs2 @ I2 @ X ) @ J2 )
        = ( ( ( I2 = J2 )
           => X )
          & ( ( I2 != J2 )
           => ( nth_o @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4476_nth__list__update,axiom,
    ! [I2: nat,Xs2: list_nat,J2: nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ( I2 = J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J2 )
            = X ) )
        & ( ( I2 != J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs2 @ I2 @ X ) @ J2 )
            = ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4477_nth__list__update,axiom,
    ! [I2: nat,Xs2: list_int,J2: nat,X: int] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ( I2 = J2 )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J2 )
            = X ) )
        & ( ( I2 != J2 )
         => ( ( nth_int @ ( list_update_int @ Xs2 @ I2 @ X ) @ J2 )
            = ( nth_int @ Xs2 @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_4478_set__encode__inf,axiom,
    ! [A4: set_nat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( nat_set_encode @ A4 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_4479_eucl__rel__int__iff,axiom,
    ! [K2: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( ( K2
          = ( plus_plus_int @ ( times_times_int @ L @ Q2 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q2 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4480_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q2: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_4481_insert__simp__excp,axiom,
    ! [Mi: nat,Deg: nat,TreeList2: list_VEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ X @ Mi )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ X @ ( ord_max_nat @ Mi @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_excp
thf(fact_4482_insert__simp__norm,axiom,
    ! [X: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
     => ( ( ord_less_nat @ Mi @ X )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( X != Ma )
           => ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( ord_max_nat @ X @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).

% insert_simp_norm
thf(fact_4483_pred__less__length__list,axiom,
    ! [Deg: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% pred_less_length_list
thf(fact_4484_pred__lesseq__max,axiom,
    ! [Deg: nat,X: nat,Ma: nat,Mi: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ X @ Ma )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% pred_lesseq_max
thf(fact_4485_succ__greatereq__min,axiom,
    ! [Deg: nat,Mi: nat,X: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            @ ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% succ_greatereq_min
thf(fact_4486_succ__less__length__list,axiom,
    ! [Deg: nat,Mi: nat,X: nat,TreeList2: list_VEBT_VEBT,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
     => ( ( ord_less_eq_nat @ Mi @ X )
       => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_option_nat
              @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
              @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ none_nat
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% succ_less_length_list
thf(fact_4487_del__in__range,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_eq_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( if_nat
                          @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            = none_nat )
                          @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                @ ( vEBT_Node
                  @ ( some_P7363390416028606310at_nat
                    @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                      @ ( if_nat
                        @ ( ( ( X = Mi )
                           => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                              = Ma ) )
                          & ( ( X != Mi )
                           => ( X = Ma ) ) )
                        @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ Summary ) )
              @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_in_range
thf(fact_4488_del__x__mi,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = 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 @ TreeList2 @ ( 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 @ TreeList2 ) )
                 => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                    = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                              @ Ma ) ) )
                        @ Deg
                        @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi
thf(fact_4489_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_4490_pred__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_pred @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ Y4 @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% pred_empty
thf(fact_4491_succ__empty,axiom,
    ! [T: vEBT_VEBT,N: nat,X: nat] :
      ( ( vEBT_invar_vebt @ T @ N )
     => ( ( ( vEBT_vebt_succ @ T @ X )
          = none_nat )
        = ( ( collect_nat
            @ ^ [Y4: nat] :
                ( ( vEBT_vebt_member @ T @ Y4 )
                & ( ord_less_nat @ X @ Y4 ) ) )
          = bot_bot_set_nat ) ) ) ).

% succ_empty
thf(fact_4492_max_Obounded__iff,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_4493_max_Obounded__iff,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
      = ( ( ord_less_eq_num @ B @ A )
        & ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_4494_max_Obounded__iff,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_4495_max_Obounded__iff,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
      = ( ( ord_less_eq_int @ B @ A )
        & ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.bounded_iff
thf(fact_4496_max_Oabsorb2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_max_rat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4497_max_Oabsorb2,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_max_num @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4498_max_Oabsorb2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_max_nat @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4499_max_Oabsorb2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_max_int @ A @ B )
        = B ) ) ).

% max.absorb2
thf(fact_4500_max_Oabsorb1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_max_rat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4501_max_Oabsorb1,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_max_num @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4502_max_Oabsorb1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_max_nat @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4503_max_Oabsorb1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_max_int @ A @ B )
        = A ) ) ).

% max.absorb1
thf(fact_4504_max__bot,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ bot_bot_set_nat @ X )
      = X ) ).

% max_bot
thf(fact_4505_max__bot,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ bot_bot_set_int @ X )
      = X ) ).

% max_bot
thf(fact_4506_max__bot,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ bot_bot_set_real @ X )
      = X ) ).

% max_bot
thf(fact_4507_max__bot,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ bot_bot_nat @ X )
      = X ) ).

% max_bot
thf(fact_4508_max__bot2,axiom,
    ! [X: set_nat] :
      ( ( ord_max_set_nat @ X @ bot_bot_set_nat )
      = X ) ).

% max_bot2
thf(fact_4509_max__bot2,axiom,
    ! [X: set_int] :
      ( ( ord_max_set_int @ X @ bot_bot_set_int )
      = X ) ).

% max_bot2
thf(fact_4510_max__bot2,axiom,
    ! [X: set_real] :
      ( ( ord_max_set_real @ X @ bot_bot_set_real )
      = X ) ).

% max_bot2
thf(fact_4511_max__bot2,axiom,
    ! [X: nat] :
      ( ( ord_max_nat @ X @ bot_bot_nat )
      = X ) ).

% max_bot2
thf(fact_4512_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_4513_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_4514_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_4515_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_4516_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_4517_max__0L,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N )
      = N ) ).

% max_0L
thf(fact_4518_max__0R,axiom,
    ! [N: nat] :
      ( ( ord_max_nat @ N @ zero_zero_nat )
      = N ) ).

% max_0R
thf(fact_4519_finite__Collect__le__nat,axiom,
    ! [K2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K2 ) ) ) ).

% finite_Collect_le_nat
thf(fact_4520_max__number__of_I1_J,axiom,
    ! [U: num,V2: num] :
      ( ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V2 ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V2 ) )
          = ( numera1916890842035813515d_enat @ V2 ) ) )
      & ( ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V2 ) )
       => ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ U ) @ ( numera1916890842035813515d_enat @ V2 ) )
          = ( numera1916890842035813515d_enat @ U ) ) ) ) ).

% max_number_of(1)
thf(fact_4521_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_4522_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_4523_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_4524_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_4525_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ zero_zero_rat @ ( numeral_numeral_rat @ X ) )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(3)
thf(fact_4526_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(3)
thf(fact_4527_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ zero_zero_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(3)
thf(fact_4528_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ zero_zero_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(3)
thf(fact_4529_max__0__1_I3_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ zero_zero_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(3)
thf(fact_4530_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_rat @ ( numeral_numeral_rat @ X ) @ zero_zero_rat )
      = ( numeral_numeral_rat @ X ) ) ).

% max_0_1(4)
thf(fact_4531_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ zero_z5237406670263579293d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(4)
thf(fact_4532_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ zero_zero_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(4)
thf(fact_4533_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ zero_zero_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(4)
thf(fact_4534_max__0__1_I4_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ zero_zero_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(4)
thf(fact_4535_max__0__1_I2_J,axiom,
    ( ( ord_max_Code_integer @ one_one_Code_integer @ zero_z3403309356797280102nteger )
    = one_one_Code_integer ) ).

% max_0_1(2)
thf(fact_4536_max__0__1_I2_J,axiom,
    ( ( ord_max_real @ one_one_real @ zero_zero_real )
    = one_one_real ) ).

% max_0_1(2)
thf(fact_4537_max__0__1_I2_J,axiom,
    ( ( ord_max_rat @ one_one_rat @ zero_zero_rat )
    = one_one_rat ) ).

% max_0_1(2)
thf(fact_4538_max__0__1_I2_J,axiom,
    ( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
    = one_one_nat ) ).

% max_0_1(2)
thf(fact_4539_max__0__1_I2_J,axiom,
    ( ( ord_max_int @ one_one_int @ zero_zero_int )
    = one_one_int ) ).

% max_0_1(2)
thf(fact_4540_max__0__1_I1_J,axiom,
    ( ( ord_max_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% max_0_1(1)
thf(fact_4541_max__0__1_I1_J,axiom,
    ( ( ord_max_real @ zero_zero_real @ one_one_real )
    = one_one_real ) ).

% max_0_1(1)
thf(fact_4542_max__0__1_I1_J,axiom,
    ( ( ord_max_rat @ zero_zero_rat @ one_one_rat )
    = one_one_rat ) ).

% max_0_1(1)
thf(fact_4543_max__0__1_I1_J,axiom,
    ( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
    = one_one_nat ) ).

% max_0_1(1)
thf(fact_4544_max__0__1_I1_J,axiom,
    ( ( ord_max_int @ zero_zero_int @ one_one_int )
    = one_one_int ) ).

% max_0_1(1)
thf(fact_4545_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ one_one_Code_integer @ ( numera6620942414471956472nteger @ X ) )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(5)
thf(fact_4546_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X ) )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(5)
thf(fact_4547_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ one_one_real @ ( numeral_numeral_real @ X ) )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(5)
thf(fact_4548_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(5)
thf(fact_4549_max__0__1_I5_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ one_one_int @ ( numeral_numeral_int @ X ) )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(5)
thf(fact_4550_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_Code_integer @ ( numera6620942414471956472nteger @ X ) @ one_one_Code_integer )
      = ( numera6620942414471956472nteger @ X ) ) ).

% max_0_1(6)
thf(fact_4551_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_ma741700101516333627d_enat @ ( numera1916890842035813515d_enat @ X ) @ one_on7984719198319812577d_enat )
      = ( numera1916890842035813515d_enat @ X ) ) ).

% max_0_1(6)
thf(fact_4552_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_real @ ( numeral_numeral_real @ X ) @ one_one_real )
      = ( numeral_numeral_real @ X ) ) ).

% max_0_1(6)
thf(fact_4553_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat )
      = ( numeral_numeral_nat @ X ) ) ).

% max_0_1(6)
thf(fact_4554_max__0__1_I6_J,axiom,
    ! [X: num] :
      ( ( ord_max_int @ ( numeral_numeral_int @ X ) @ one_one_int )
      = ( numeral_numeral_int @ X ) ) ).

% max_0_1(6)
thf(fact_4555_card__Collect__le__nat,axiom,
    ! [N: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I: nat] : ( ord_less_eq_nat @ I @ N ) ) )
      = ( suc @ N ) ) ).

% card_Collect_le_nat
thf(fact_4556_del__x__not__mia,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
               => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                  = ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
                    @ ( vEBT_Node
                      @ ( some_P7363390416028606310at_nat
                        @ ( product_Pair_nat_nat @ Mi
                          @ ( if_nat @ ( X = 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ H ) ) ) ) ) ) ) )
                            @ Ma ) ) )
                      @ Deg
                      @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
                      @ ( vEBT_vebt_delete @ Summary @ H ) )
                    @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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 @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ H ) ) ) ) @ Ma ) ) ) @ Deg @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) ) @ Summary ) ) ) ) ) ) ) ) ) ).

% del_x_not_mia
thf(fact_4557_del__x__not__mi__new__node__nil,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Sn: vEBT_VEBT,Summary: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
               => ( ( vEBT_VEBT_minNull @ Newnode )
                 => ( ( Sn
                      = ( vEBT_vebt_delete @ Summary @ H ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ Newnode ) )
                     => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ Sn )
                                      = none_nat )
                                    @ Mi
                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                  @ Ma ) ) )
                            @ Deg
                            @ Newlist
                            @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_not_mi_new_node_nil
thf(fact_4558_del__x__not__mi,axiom,
    ! [Mi: nat,X: nat,Ma: nat,Deg: nat,H: nat,L: nat,Newnode: vEBT_VEBT,TreeList2: list_VEBT_VEBT,Newlist: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Mi @ X )
        & ( ord_less_eq_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = H )
           => ( ( ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
                = L )
             => ( ( Newnode
                  = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
               => ( ( Newlist
                    = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ Newnode ) )
                 => ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                   => ( ( ( vEBT_VEBT_minNull @ Newnode )
                       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node
                            @ ( some_P7363390416028606310at_nat
                              @ ( product_Pair_nat_nat @ Mi
                                @ ( if_nat @ ( X = Ma )
                                  @ ( if_nat
                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ 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 @ TreeList2 @ Summary ) @ X )
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ ( if_nat @ ( X = 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_4559_del__x__mia,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
            = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( 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 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                        @ Ma ) ) )
                  @ Deg
                  @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ ( 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 @ TreeList2 @ Summary ) ) ) ) ) ) ).

% del_x_mia
thf(fact_4560_del__x__mi__lets__in__minNull,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT,Sn: vEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = 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 @ TreeList2 @ ( 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 @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ 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 @ TreeList2 @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ Sn )
                                        = none_nat )
                                      @ Xn
                                      @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ Newlist @ ( the_nat @ ( vEBT_vebt_maxt @ Sn ) ) ) ) ) ) )
                                    @ Ma ) ) )
                              @ Deg
                              @ Newlist
                              @ Sn ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% del_x_mi_lets_in_minNull
thf(fact_4561_del__x__mi__lets__in,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Deg: nat,Xn: nat,H: nat,Summary: vEBT_VEBT,TreeList2: list_VEBT_VEBT,L: nat,Newnode: vEBT_VEBT,Newlist: list_VEBT_VEBT] :
      ( ( ( X = Mi )
        & ( ord_less_nat @ X @ Ma ) )
     => ( ( Mi != Ma )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
         => ( ( ( vEBT_VEBT_high @ Xn @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
              = 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 @ TreeList2 @ ( 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 @ TreeList2 ) )
                 => ( ( Newnode
                      = ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ H ) @ L ) )
                   => ( ( Newlist
                        = ( list_u1324408373059187874T_VEBT @ TreeList2 @ H @ Newnode ) )
                     => ( ( ( vEBT_VEBT_minNull @ Newnode )
                         => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X )
                            = ( vEBT_Node
                              @ ( some_P7363390416028606310at_nat
                                @ ( product_Pair_nat_nat @ Xn
                                  @ ( if_nat @ ( Xn = Ma )
                                    @ ( if_nat
                                      @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ 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 @ TreeList2 @ Summary ) @ X )
                            = ( 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_4562_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri5074537144036343181t_real @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_real @ ( semiri5074537144036343181t_real @ X ) @ ( semiri5074537144036343181t_real @ Y ) ) ) ).

% of_nat_max
thf(fact_4563_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).

% of_nat_max
thf(fact_4564_of__nat__max,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X @ Y ) )
      = ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).

% of_nat_max
thf(fact_4565_max__def__raw,axiom,
    ( ord_max_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4566_max__def__raw,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4567_max__def__raw,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4568_max__def__raw,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4569_max__def__raw,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def_raw
thf(fact_4570_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_4571_lambda__zero,axiom,
    ( ( ^ [H2: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_4572_lambda__zero,axiom,
    ( ( ^ [H2: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_4573_lambda__zero,axiom,
    ( ( ^ [H2: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_4574_lambda__zero,axiom,
    ( ( ^ [H2: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_4575_lambda__zero,axiom,
    ( ( ^ [H2: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_4576_lambda__one,axiom,
    ( ( ^ [X4: code_integer] : X4 )
    = ( times_3573771949741848930nteger @ one_one_Code_integer ) ) ).

% lambda_one
thf(fact_4577_lambda__one,axiom,
    ( ( ^ [X4: complex] : X4 )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_4578_lambda__one,axiom,
    ( ( ^ [X4: real] : X4 )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_4579_lambda__one,axiom,
    ( ( ^ [X4: rat] : X4 )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_4580_lambda__one,axiom,
    ( ( ^ [X4: nat] : X4 )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_4581_lambda__one,axiom,
    ( ( ^ [X4: int] : X4 )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_4582_finite__M__bounded__by__nat,axiom,
    ! [P2: nat > $o,I2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P2 @ K3 )
            & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_4583_finite__less__ub,axiom,
    ! [F2: nat > nat,U: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F2 @ N3 ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [N4: nat] : ( ord_less_eq_nat @ ( F2 @ N4 ) @ U ) ) ) ) ).

% finite_less_ub
thf(fact_4584_max_Omono,axiom,
    ! [C2: rat,A: rat,D: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ( ord_less_eq_rat @ D @ B )
       => ( ord_less_eq_rat @ ( ord_max_rat @ C2 @ D ) @ ( ord_max_rat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4585_max_Omono,axiom,
    ! [C2: num,A: num,D: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ( ord_less_eq_num @ D @ B )
       => ( ord_less_eq_num @ ( ord_max_num @ C2 @ D ) @ ( ord_max_num @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4586_max_Omono,axiom,
    ! [C2: nat,A: nat,D: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ( ord_less_eq_nat @ D @ B )
       => ( ord_less_eq_nat @ ( ord_max_nat @ C2 @ D ) @ ( ord_max_nat @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4587_max_Omono,axiom,
    ! [C2: int,A: int,D: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ( ord_less_eq_int @ D @ B )
       => ( ord_less_eq_int @ ( ord_max_int @ C2 @ D ) @ ( ord_max_int @ A @ B ) ) ) ) ).

% max.mono
thf(fact_4588_max_OorderE,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( A
        = ( ord_max_rat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4589_max_OorderE,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( A
        = ( ord_max_num @ A @ B ) ) ) ).

% max.orderE
thf(fact_4590_max_OorderE,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( A
        = ( ord_max_nat @ A @ B ) ) ) ).

% max.orderE
thf(fact_4591_max_OorderE,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( A
        = ( ord_max_int @ A @ B ) ) ) ).

% max.orderE
thf(fact_4592_max_OorderI,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( ord_max_rat @ A @ B ) )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% max.orderI
thf(fact_4593_max_OorderI,axiom,
    ! [A: num,B: num] :
      ( ( A
        = ( ord_max_num @ A @ B ) )
     => ( ord_less_eq_num @ B @ A ) ) ).

% max.orderI
thf(fact_4594_max_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( ord_max_nat @ A @ B ) )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% max.orderI
thf(fact_4595_max_OorderI,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( ord_max_int @ A @ B ) )
     => ( ord_less_eq_int @ B @ A ) ) ).

% max.orderI
thf(fact_4596_max_OboundedE,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_rat @ B @ A )
         => ~ ( ord_less_eq_rat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_4597_max_OboundedE,axiom,
    ! [B: num,C2: num,A: num] :
      ( ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_num @ B @ A )
         => ~ ( ord_less_eq_num @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_4598_max_OboundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_nat @ B @ A )
         => ~ ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_4599_max_OboundedE,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A )
     => ~ ( ( ord_less_eq_int @ B @ A )
         => ~ ( ord_less_eq_int @ C2 @ A ) ) ) ).

% max.boundedE
thf(fact_4600_max_OboundedI,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C2 @ A )
       => ( ord_less_eq_rat @ ( ord_max_rat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_4601_max_OboundedI,axiom,
    ! [B: num,A: num,C2: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C2 @ A )
       => ( ord_less_eq_num @ ( ord_max_num @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_4602_max_OboundedI,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ A )
       => ( ord_less_eq_nat @ ( ord_max_nat @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_4603_max_OboundedI,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C2 @ A )
       => ( ord_less_eq_int @ ( ord_max_int @ B @ C2 ) @ A ) ) ) ).

% max.boundedI
thf(fact_4604_max_Oorder__iff,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( A5
          = ( ord_max_rat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4605_max_Oorder__iff,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( A5
          = ( ord_max_num @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4606_max_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( A5
          = ( ord_max_nat @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4607_max_Oorder__iff,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( A5
          = ( ord_max_int @ A5 @ B4 ) ) ) ) ).

% max.order_iff
thf(fact_4608_max_Ocobounded1,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ A @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4609_max_Ocobounded1,axiom,
    ! [A: num,B: num] : ( ord_less_eq_num @ A @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded1
thf(fact_4610_max_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded1
thf(fact_4611_max_Ocobounded1,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ A @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded1
thf(fact_4612_max_Ocobounded2,axiom,
    ! [B: rat,A: rat] : ( ord_less_eq_rat @ B @ ( ord_max_rat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4613_max_Ocobounded2,axiom,
    ! [B: num,A: num] : ( ord_less_eq_num @ B @ ( ord_max_num @ A @ B ) ) ).

% max.cobounded2
thf(fact_4614_max_Ocobounded2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( ord_max_nat @ A @ B ) ) ).

% max.cobounded2
thf(fact_4615_max_Ocobounded2,axiom,
    ! [B: int,A: int] : ( ord_less_eq_int @ B @ ( ord_max_int @ A @ B ) ) ).

% max.cobounded2
thf(fact_4616_le__max__iff__disj,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ Z3 @ ( ord_max_rat @ X @ Y ) )
      = ( ( ord_less_eq_rat @ Z3 @ X )
        | ( ord_less_eq_rat @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4617_le__max__iff__disj,axiom,
    ! [Z3: num,X: num,Y: num] :
      ( ( ord_less_eq_num @ Z3 @ ( ord_max_num @ X @ Y ) )
      = ( ( ord_less_eq_num @ Z3 @ X )
        | ( ord_less_eq_num @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4618_le__max__iff__disj,axiom,
    ! [Z3: nat,X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ Z3 @ ( ord_max_nat @ X @ Y ) )
      = ( ( ord_less_eq_nat @ Z3 @ X )
        | ( ord_less_eq_nat @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4619_le__max__iff__disj,axiom,
    ! [Z3: int,X: int,Y: int] :
      ( ( ord_less_eq_int @ Z3 @ ( ord_max_int @ X @ Y ) )
      = ( ( ord_less_eq_int @ Z3 @ X )
        | ( ord_less_eq_int @ Z3 @ Y ) ) ) ).

% le_max_iff_disj
thf(fact_4620_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_rat
    = ( ^ [B4: rat,A5: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4621_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_num
    = ( ^ [B4: num,A5: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4622_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4623_max_Oabsorb__iff1,axiom,
    ( ord_less_eq_int
    = ( ^ [B4: int,A5: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = A5 ) ) ) ).

% max.absorb_iff1
thf(fact_4624_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( ord_max_rat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4625_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_num
    = ( ^ [A5: num,B4: num] :
          ( ( ord_max_num @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4626_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_max_nat @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4627_max_Oabsorb__iff2,axiom,
    ( ord_less_eq_int
    = ( ^ [A5: int,B4: int] :
          ( ( ord_max_int @ A5 @ B4 )
          = B4 ) ) ) ).

% max.absorb_iff2
thf(fact_4628_max_OcoboundedI1,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C2 @ A )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4629_max_OcoboundedI1,axiom,
    ! [C2: num,A: num,B: num] :
      ( ( ord_less_eq_num @ C2 @ A )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4630_max_OcoboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ A )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4631_max_OcoboundedI1,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C2 @ A )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI1
thf(fact_4632_max_OcoboundedI2,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ C2 @ B )
     => ( ord_less_eq_rat @ C2 @ ( ord_max_rat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4633_max_OcoboundedI2,axiom,
    ! [C2: num,B: num,A: num] :
      ( ( ord_less_eq_num @ C2 @ B )
     => ( ord_less_eq_num @ C2 @ ( ord_max_num @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4634_max_OcoboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ C2 @ B )
     => ( ord_less_eq_nat @ C2 @ ( ord_max_nat @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4635_max_OcoboundedI2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( ord_less_eq_int @ C2 @ B )
     => ( ord_less_eq_int @ C2 @ ( ord_max_int @ A @ B ) ) ) ).

% max.coboundedI2
thf(fact_4636_max__absorb2,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ( ord_max_set_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4637_max__absorb2,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X @ Y )
     => ( ( ord_max_rat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4638_max__absorb2,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_num @ X @ Y )
     => ( ( ord_max_num @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4639_max__absorb2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_max_nat @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4640_max__absorb2,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ( ord_max_int @ X @ Y )
        = Y ) ) ).

% max_absorb2
thf(fact_4641_max__absorb1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ X )
     => ( ( ord_max_set_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4642_max__absorb1,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ Y @ X )
     => ( ( ord_max_rat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4643_max__absorb1,axiom,
    ! [Y: num,X: num] :
      ( ( ord_less_eq_num @ Y @ X )
     => ( ( ord_max_num @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4644_max__absorb1,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_max_nat @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4645_max__absorb1,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ Y @ X )
     => ( ( ord_max_int @ X @ Y )
        = X ) ) ).

% max_absorb1
thf(fact_4646_max__def,axiom,
    ( ord_max_set_nat
    = ( ^ [A5: set_nat,B4: set_nat] : ( if_set_nat @ ( ord_less_eq_set_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_4647_max__def,axiom,
    ( ord_max_rat
    = ( ^ [A5: rat,B4: rat] : ( if_rat @ ( ord_less_eq_rat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_4648_max__def,axiom,
    ( ord_max_num
    = ( ^ [A5: num,B4: num] : ( if_num @ ( ord_less_eq_num @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_4649_max__def,axiom,
    ( ord_max_nat
    = ( ^ [A5: nat,B4: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_4650_max__def,axiom,
    ( ord_max_int
    = ( ^ [A5: int,B4: int] : ( if_int @ ( ord_less_eq_int @ A5 @ B4 ) @ B4 @ A5 ) ) ) ).

% max_def
thf(fact_4651_max__add__distrib__left,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( plus_plus_real @ ( ord_max_real @ X @ Y ) @ Z3 )
      = ( ord_max_real @ ( plus_plus_real @ X @ Z3 ) @ ( plus_plus_real @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_4652_max__add__distrib__left,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( plus_plus_rat @ ( ord_max_rat @ X @ Y ) @ Z3 )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Z3 ) @ ( plus_plus_rat @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_4653_max__add__distrib__left,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ X @ Y ) @ Z3 )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Z3 ) @ ( plus_plus_nat @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_4654_max__add__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ord_max_int @ ( plus_plus_int @ X @ Z3 ) @ ( plus_plus_int @ Y @ Z3 ) ) ) ).

% max_add_distrib_left
thf(fact_4655_max__add__distrib__right,axiom,
    ! [X: real,Y: real,Z3: real] :
      ( ( plus_plus_real @ X @ ( ord_max_real @ Y @ Z3 ) )
      = ( ord_max_real @ ( plus_plus_real @ X @ Y ) @ ( plus_plus_real @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_4656_max__add__distrib__right,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( plus_plus_rat @ X @ ( ord_max_rat @ Y @ Z3 ) )
      = ( ord_max_rat @ ( plus_plus_rat @ X @ Y ) @ ( plus_plus_rat @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_4657_max__add__distrib__right,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( plus_plus_nat @ X @ ( ord_max_nat @ Y @ Z3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ X @ Y ) @ ( plus_plus_nat @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_4658_max__add__distrib__right,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( plus_plus_int @ X @ ( ord_max_int @ Y @ Z3 ) )
      = ( ord_max_int @ ( plus_plus_int @ X @ Y ) @ ( plus_plus_int @ X @ Z3 ) ) ) ).

% max_add_distrib_right
thf(fact_4659_max__diff__distrib__left,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( minus_minus_rat @ ( ord_max_rat @ X @ Y ) @ Z3 )
      = ( ord_max_rat @ ( minus_minus_rat @ X @ Z3 ) @ ( minus_minus_rat @ Y @ Z3 ) ) ) ).

% max_diff_distrib_left
thf(fact_4660_max__diff__distrib__left,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( minus_minus_int @ ( ord_max_int @ X @ Y ) @ Z3 )
      = ( ord_max_int @ ( minus_minus_int @ X @ Z3 ) @ ( minus_minus_int @ Y @ Z3 ) ) ) ).

% max_diff_distrib_left
thf(fact_4661_nat__add__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N ) @ Q2 )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q2 ) @ ( plus_plus_nat @ N @ Q2 ) ) ) ).

% nat_add_max_left
thf(fact_4662_nat__add__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M2 @ N ) @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ).

% nat_add_max_right
thf(fact_4663_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_4664_nat__mult__max__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M2 @ N ) @ Q2 )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_max_left
thf(fact_4665_nat__mult__max__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M2 @ ( ord_max_nat @ N @ Q2 ) )
      = ( ord_max_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q2 ) ) ) ).

% nat_mult_max_right
thf(fact_4666_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_4667_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) ) ).

% numeral_code(2)
thf(fact_4668_numeral__code_I2_J,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_code(2)
thf(fact_4669_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_4670_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_4671_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_4672_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).

% nat_leq_as_int
thf(fact_4673_power__numeral__even,axiom,
    ! [Z3: complex,W2: num] :
      ( ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_4674_power__numeral__even,axiom,
    ! [Z3: real,W2: num] :
      ( ( power_power_real @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_real @ ( power_power_real @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_real @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_4675_power__numeral__even,axiom,
    ! [Z3: rat,W2: num] :
      ( ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_rat @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_4676_power__numeral__even,axiom,
    ! [Z3: nat,W2: num] :
      ( ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_4677_power__numeral__even,axiom,
    ! [Z3: int,W2: num] :
      ( ( power_power_int @ Z3 @ ( numeral_numeral_nat @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_even
thf(fact_4678_card__less,axiom,
    ! [M5: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_4679_card__less__Suc,axiom,
    ! [M5: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M5 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M5 )
                  & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_4680_card__less__Suc2,axiom,
    ! [M5: set_nat,I2: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M5 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M5 )
                & ( ord_less_nat @ K3 @ I2 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M5 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_4681_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_4682_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_4683_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_4684_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_4685_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_4686_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_4687_card__lists__length__eq,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A4 )
                & ( ( size_s3023201423986296836st_nat @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_list_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4688_card__lists__length__eq,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
                & ( ( size_s3254054031482475050et_nat @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_set_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4689_card__lists__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
                & ( ( size_s3451745648224563538omplex @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_complex @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4690_card__lists__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
                & ( ( size_s6755466524823107622T_VEBT @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4691_card__lists__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_card_list_o
          @ ( collect_list_o
            @ ^ [Xs: list_o] :
                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 )
                & ( ( size_size_list_o @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_o @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4692_card__lists__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs: list_int] :
                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 )
                & ( ( size_size_list_int @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_int @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4693_card__lists__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs: list_nat] :
                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
                & ( ( size_size_list_nat @ Xs )
                  = N ) ) ) )
        = ( power_power_nat @ ( finite_card_nat @ A4 ) @ N ) ) ) ).

% card_lists_length_eq
thf(fact_4694_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_4695_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_4696_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_4697_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_4698_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_4699_vebt__insert_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
      = ( if_VEBT_VEBT
        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
          & ~ ( ( X = Mi )
              | ( X = Ma ) ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( 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 @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary ) )
        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) ) ).

% vebt_insert.simps(5)
thf(fact_4700_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList2: list_VEBT_VEBT,S2: vEBT_VEBT,X: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList2 @ S2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_4701_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V2: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList2 @ Vd2 ) @ X )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_4702_vebt__insert_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( 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 ) ) ) ) ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) )
             => ( Y
               != ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
               => ( Y
                 != ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) ) )
           => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                   => ( Y
                     != ( if_VEBT_VEBT
                        @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                          & ~ ( ( Xa2 = Mi2 )
                              | ( Xa2 = Ma2 ) ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ).

% vebt_insert.elims
thf(fact_4703_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
      = ( ( X != Mi )
       => ( ( X != Ma )
         => ( ~ ( ord_less_nat @ X @ Mi )
            & ( ~ ( ord_less_nat @ X @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X )
                & ( ~ ( ord_less_nat @ Ma @ X )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_4704_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V2: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList2 @ Vc ) @ X )
      = ( ( X = Mi )
        | ( X = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_4705_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_4706_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ~ ( ( ( Xa2 = zero_zero_nat )
                 => A3 )
                & ( ( Xa2 != zero_zero_nat )
                 => ( ( ( Xa2 = one_one_nat )
                     => B3 )
                    & ( Xa2 = one_one_nat ) ) ) ) )
       => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
              ( ? [S: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_4707_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) )
           => Y )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [S: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_4708_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X
                = ( 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,TreeList: list_VEBT_VEBT] :
              ( ? [Vc2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
         => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_4709_vebt__member_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( 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,Va: nat,TreeList: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
             => ~ ( ( Xa2 != Mi2 )
                 => ( ( Xa2 != Ma2 )
                   => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                      & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                       => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                          & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                           => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                               => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_4710_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( 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,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_4711_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
           => Y )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X
                    = ( 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,TreeList: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) )
             => ~ ! [V: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_4712_vebt__member_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( ( Xa2 = zero_zero_nat )
               => A3 )
              & ( ( Xa2 != zero_zero_nat )
               => ( ( ( Xa2 = one_one_nat )
                   => B3 )
                  & ( Xa2 = one_one_nat ) ) ) ) )
       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
              ( X
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
         => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
           => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                   => ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_4713_vebt__member_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
              = ( ~ ( ( ( Xa2 = zero_zero_nat )
                     => A3 )
                    & ( ( Xa2 != zero_zero_nat )
                     => ( ( ( Xa2 = one_one_nat )
                         => B3 )
                        & ( Xa2 = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( X
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
           => Y )
         => ( ( ? [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
             => Y )
           => ( ( ? [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y )
             => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                   => ( Y
                      = ( ~ ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_4714_vebt__pred_Osimps_I7_J,axiom,
    ! [Ma: nat,X: nat,Mi: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Ma ) ) )
      & ( ~ ( ord_less_nat @ Ma @ X )
       => ( ( vEBT_vebt_pred @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  = none_nat )
                @ ( if_option_nat @ ( ord_less_nat @ Mi @ X ) @ ( some_nat @ Mi ) @ none_nat )
                @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_pred @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_pred.simps(7)
thf(fact_4715_vebt__succ_Osimps_I6_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
          = ( some_nat @ Mi ) ) )
      & ( ~ ( ord_less_nat @ X @ Mi )
       => ( ( vEBT_vebt_succ @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
          = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                 != none_nat )
                & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( 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 @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
              @ ( if_option_nat
                @ ( ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( 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 @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_succ @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
            @ none_nat ) ) ) ) ).

% vebt_succ.simps(6)
thf(fact_4716_vebt__delete_Osimps_I7_J,axiom,
    ! [X: nat,Mi: nat,Ma: nat,Va3: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( ( ( ord_less_nat @ X @ Mi )
          | ( ord_less_nat @ Ma @ X ) )
       => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) )
      & ( ~ ( ( ord_less_nat @ X @ Mi )
            | ( ord_less_nat @ Ma @ X ) )
       => ( ( ( ( X = Mi )
              & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
              = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) )
          & ( ~ ( ( X = Mi )
                & ( X = Ma ) )
           => ( ( vEBT_vebt_delete @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) @ X )
              = ( if_VEBT_VEBT @ ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( if_nat
                            @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              = none_nat )
                            @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                            @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va3 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ ( vEBT_vebt_delete @ Summary @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  @ ( vEBT_Node
                    @ ( some_P7363390416028606310at_nat
                      @ ( product_Pair_nat_nat @ ( if_nat @ ( X = Mi ) @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ Mi )
                        @ ( if_nat
                          @ ( ( ( X = Mi )
                             => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) )
                                = Ma ) )
                            & ( ( X != Mi )
                             => ( X = Ma ) ) )
                          @ ( plus_plus_nat @ ( times_times_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( 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 @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                          @ Ma ) ) )
                    @ ( suc @ ( suc @ Va3 ) )
                    @ ( list_u1324408373059187874T_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( X = Mi ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList2 @ ( the_nat @ ( vEBT_vebt_mint @ Summary ) ) ) ) ) ) @ X ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    @ Summary ) )
                @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va3 ) ) @ TreeList2 @ Summary ) ) ) ) ) ) ) ).

% vebt_delete.simps(7)
thf(fact_4717_vebt__delete_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ( Y
               != ( vEBT_Leaf @ $false @ B3 ) ) ) )
       => ( ! [A3: $o] :
              ( ? [B3: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ( Y
                 != ( vEBT_Leaf @ A3 @ $false ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [N3: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ N3 ) ) )
                 => ( Y
                   != ( vEBT_Leaf @ A3 @ B3 ) ) ) )
           => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                 => ( Y
                   != ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) ) )
             => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                   => ( Y
                     != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) ) )
               => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                     => ( Y
                       != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( Y
                                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                            & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                             => ( ( ( ( Xa2 = Mi2 )
                                    & ( Xa2 = Ma2 ) )
                                 => ( Y
                                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                                & ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = Ma2 ) )
                                                  & ( ( Xa2 != Mi2 )
                                                   => ( Xa2 = Ma2 ) ) )
                                                @ ( if_nat
                                                  @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                    = none_nat )
                                                  @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                  @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        @ ( vEBT_Node
                                          @ ( some_P7363390416028606310at_nat
                                            @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                              @ ( if_nat
                                                @ ( ( ( Xa2 = Mi2 )
                                                   => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                      = 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                @ Ma2 ) ) )
                                          @ ( suc @ ( suc @ Va ) )
                                          @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ Summary2 ) )
                                      @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.elims
thf(fact_4718_vebt__succ_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ! [Uu2: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ Uu2 @ B3 ) )
           => ( ( Xa2 = zero_zero_nat )
             => ~ ( ( B3
                   => ( Y
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B3
                   => ( Y = none_nat ) ) ) ) )
       => ( ( ? [Uv2: $o,Uw: $o] :
                ( X
                = ( vEBT_Leaf @ Uv2 @ Uw ) )
           => ( ? [N3: nat] :
                  ( Xa2
                  = ( suc @ N3 ) )
             => ( Y != none_nat ) ) )
         => ( ( ? [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
             => ( Y != none_nat ) )
           => ( ( ? [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                 => ( Y != none_nat ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ~ ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( some_nat @ Mi2 ) ) )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( Y
                              = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                @ ( if_option_nat
                                  @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                     != none_nat )
                                    & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                  @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  @ ( if_option_nat
                                    @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                      = none_nat )
                                    @ none_nat
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                @ none_nat ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.elims
thf(fact_4719_vebt__pred_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( ( Xa2 = zero_zero_nat )
           => ( Y != none_nat ) ) )
       => ( ! [A3: $o] :
              ( ? [Uw: $o] :
                  ( X
                  = ( vEBT_Leaf @ A3 @ Uw ) )
             => ( ( Xa2
                  = ( suc @ zero_zero_nat ) )
               => ~ ( ( A3
                     => ( Y
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A3
                     => ( Y = none_nat ) ) ) ) )
         => ( ! [A3: $o,B3: $o] :
                ( ( X
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ( ? [Va: nat] :
                      ( Xa2
                      = ( suc @ ( suc @ Va ) ) )
                 => ~ ( ( B3
                       => ( Y
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B3
                       => ( ( A3
                           => ( Y
                              = ( some_nat @ zero_zero_nat ) ) )
                          & ( ~ A3
                           => ( Y = none_nat ) ) ) ) ) ) )
           => ( ( ? [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va2 ) )
               => ( Y != none_nat ) )
             => ( ( ? [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                 => ( Y != none_nat ) )
               => ( ( ? [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                   => ( Y != none_nat ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ~ ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( some_nat @ Ma2 ) ) )
                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.elims
thf(fact_4720_finite__nth__roots,axiom,
    ! [N: nat,C2: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z4: complex] :
              ( ( power_power_complex @ Z4 @ N )
              = C2 ) ) ) ) ).

% finite_nth_roots
thf(fact_4721_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_real
          @ ( collect_real
            @ ^ [Z4: real] :
                ( ( power_power_real @ Z4 @ N )
                = one_one_real ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_4722_card__roots__unity,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ord_less_eq_nat
        @ ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) ) )
        @ N ) ) ).

% card_roots_unity
thf(fact_4723_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_4724_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_4725_vebt__succ_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_succ @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ 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 @ Uu2 @ B3 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [Uv2: $o,Uw: $o] :
                ( ( X
                  = ( vEBT_Leaf @ Uv2 @ Uw ) )
               => ! [N3: nat] :
                    ( ( Xa2
                      = ( suc @ N3 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uv2 @ Uw ) @ ( suc @ N3 ) ) ) ) ) )
           => ( ! [Ux: nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) )
                 => ( ( Y = none_nat )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Ux @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vc2: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vc2 @ Vd ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vg: list_VEBT_VEBT,Vh: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vg @ Vh ) @ Xa2 ) ) ) )
                 => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                       => ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( some_nat @ Mi2 ) ) )
                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                             => ( Y
                                = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                  @ ( if_option_nat
                                    @ ( ( ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                       != none_nat )
                                      & ( vEBT_VEBT_less @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                    @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_succ @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                    @ ( if_option_nat
                                      @ ( ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                        = none_nat )
                                      @ none_nat
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_succ @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                  @ none_nat ) ) ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_succ_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_succ.pelims
thf(fact_4726_vebt__pred_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: option_nat] :
      ( ( ( vEBT_vebt_pred @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Xa2 = zero_zero_nat )
               => ( ( Y = none_nat )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ zero_zero_nat ) ) ) ) )
         => ( ! [A3: $o,Uw: $o] :
                ( ( X
                  = ( 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] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [Va: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ Va ) ) )
                     => ( ( ( 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 @ Va ) ) ) ) ) ) )
             => ( ! [Uy: nat,Uz2: list_VEBT_VEBT,Va2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va2 ) )
                   => ( ( Y = none_nat )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uy @ Uz2 @ Va2 ) @ Xa2 ) ) ) )
               => ( ! [V: product_prod_nat_nat,Vd: list_VEBT_VEBT,Ve: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) )
                     => ( ( Y = none_nat )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Vd @ Ve ) @ Xa2 ) ) ) )
                 => ( ! [V: product_prod_nat_nat,Vh: list_VEBT_VEBT,Vi: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) )
                       => ( ( Y = none_nat )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vh @ Vi ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( some_nat @ Ma2 ) ) )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( Y
                                  = ( if_option_nat @ ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                    @ ( if_option_nat
                                      @ ( ( ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                         != none_nat )
                                        & ( vEBT_VEBT_greater @ ( some_nat @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                      @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( some_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_pred @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      @ ( if_option_nat
                                        @ ( ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                                          = none_nat )
                                        @ ( if_option_nat @ ( ord_less_nat @ Mi2 @ Xa2 ) @ ( some_nat @ Mi2 ) @ none_nat )
                                        @ ( vEBT_VEBT_add @ ( vEBT_VEBT_mul @ ( some_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_pred @ Summary2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
                                    @ none_nat ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_pred_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_pred.pelims
thf(fact_4727_vebt__delete_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_delete @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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] :
                ( ( X
                  = ( 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] :
                  ( ( X
                    = ( vEBT_Leaf @ A3 @ B3 ) )
                 => ! [N3: nat] :
                      ( ( Xa2
                        = ( suc @ ( suc @ N3 ) ) )
                     => ( ( Y
                          = ( vEBT_Leaf @ A3 @ B3 ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ ( suc @ ( suc @ N3 ) ) ) ) ) ) )
             => ( ! [Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
               => ( ! [Mi2: nat,Ma2: nat,TrLst: list_VEBT_VEBT,Smry: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                     => ( ( Y
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ TrLst @ Smry ) @ Xa2 ) ) ) )
                 => ( ! [Mi2: nat,Ma2: nat,Tr: list_VEBT_VEBT,Sm: vEBT_VEBT] :
                        ( ( X
                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                       => ( ( Y
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) )
                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ zero_zero_nat ) @ Tr @ Sm ) @ Xa2 ) ) ) )
                   => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                          ( ( X
                            = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                         => ( ( ( ( ( ord_less_nat @ Xa2 @ Mi2 )
                                  | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( Y
                                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                              & ( ~ ( ( ord_less_nat @ Xa2 @ Mi2 )
                                    | ( ord_less_nat @ Ma2 @ Xa2 ) )
                               => ( ( ( ( Xa2 = Mi2 )
                                      & ( Xa2 = Ma2 ) )
                                   => ( Y
                                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                                  & ( ~ ( ( 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = Ma2 ) )
                                                    & ( ( Xa2 != Mi2 )
                                                     => ( Xa2 = Ma2 ) ) )
                                                  @ ( if_nat
                                                    @ ( ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                                      = none_nat )
                                                    @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                    @ ( plus_plus_nat @ ( times_times_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( 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 @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ ( vEBT_vebt_delete @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                          @ ( vEBT_Node
                                            @ ( some_P7363390416028606310at_nat
                                              @ ( product_Pair_nat_nat @ ( if_nat @ ( Xa2 = Mi2 ) @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ Mi2 )
                                                @ ( if_nat
                                                  @ ( ( ( Xa2 = Mi2 )
                                                     => ( ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) )
                                                        = 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 @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_maxt @ ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) )
                                                  @ Ma2 ) ) )
                                            @ ( suc @ ( suc @ Va ) )
                                            @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_delete @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( the_nat @ ( vEBT_vebt_mint @ ( nth_VEBT_VEBT @ TreeList @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( Xa2 = Mi2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( the_nat @ ( vEBT_vebt_mint @ Summary2 ) ) @ ( 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 @ Summary2 ) ) ) ) ) ) @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                            @ Summary2 ) )
                                        @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) ) ) ) ) )
                           => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_delete_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_delete.pelims
thf(fact_4728_vebt__insert_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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 ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) )
               => ( ( Y
                    = ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ zero_zero_nat @ Ts @ S ) @ Xa2 ) ) ) )
           => ( ! [Info: option4927543243414619207at_nat,Ts: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                 => ( ( Y
                      = ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ ( suc @ zero_zero_nat ) @ Ts @ S ) @ Xa2 ) ) ) )
             => ( ! [V: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) )
                   => ( ( Y
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
               => ~ ! [Mi2: nat,Ma2: nat,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ( ( Y
                          = ( if_VEBT_VEBT
                            @ ( ( ord_less_nat @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                              & ~ ( ( Xa2 = Mi2 )
                                  | ( Xa2 = Ma2 ) ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max_nat @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_u1324408373059187874T_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_insert @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( if_VEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if_nat @ ( ord_less_nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ Summary2 ) )
                            @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_insert.pelims
thf(fact_4729_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X4: nat,N4: nat] : ( modulo_modulo_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% low_def
thf(fact_4730_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_4731_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_4732_mod__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% mod_0
thf(fact_4733_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_4734_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_4735_mod__by__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ zero_z2226904508553997617atural )
      = A ) ).

% mod_by_0
thf(fact_4736_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_4737_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_4738_mod__self,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ A )
      = zero_z2226904508553997617atural ) ).

% mod_self
thf(fact_4739_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_4740_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_4741_bits__mod__0,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ zero_z2226904508553997617atural @ A )
      = zero_z2226904508553997617atural ) ).

% bits_mod_0
thf(fact_4742_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_4743_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_4744_mod__mult__self2__is__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self2_is_0
thf(fact_4745_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_4746_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_4747_mod__mult__self1__is__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ B @ A ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_mult_self1_is_0
thf(fact_4748_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_4749_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_4750_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_4751_mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% mod_by_1
thf(fact_4752_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_4753_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_4754_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_4755_bits__mod__by__1,axiom,
    ! [A: code_natural] :
      ( ( modulo8411746178871703098atural @ A @ one_one_Code_natural )
      = zero_z2226904508553997617atural ) ).

% bits_mod_by_1
thf(fact_4756_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_4757_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_4758_mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% mod_div_trivial
thf(fact_4759_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_4760_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_4761_bits__mod__div__trivial,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( divide5121882707175180666atural @ ( modulo8411746178871703098atural @ A @ B ) @ B )
      = zero_z2226904508553997617atural ) ).

% bits_mod_div_trivial
thf(fact_4762_mod__mult__self1,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C2 @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4763_mod__mult__self1,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C2 @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4764_mod__mult__self1,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ C2 @ B ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self1
thf(fact_4765_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C2 ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4766_mod__mult__self2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C2 ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4767_mod__mult__self2,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ A @ ( times_2397367101498566445atural @ B @ C2 ) ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self2
thf(fact_4768_mod__mult__self3,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C2 @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4769_mod__mult__self3,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C2 @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4770_mod__mult__self3,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ C2 @ B ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self3
thf(fact_4771_mod__mult__self4,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C2 ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4772_mod__mult__self4,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C2 ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4773_mod__mult__self4,axiom,
    ! [B: code_natural,C2: code_natural,A: code_natural] :
      ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ C2 ) @ A ) @ B )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% mod_mult_self4
thf(fact_4774_mod__by__Suc__0,axiom,
    ! [M2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_4775_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_4776_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_4777_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_4778_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_4779_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_4780_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_4781_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_4782_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_4783_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_4784_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != zero_z2226904508553997617atural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = one_one_Code_natural ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_4785_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_4786_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_4787_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_4788_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
       != one_one_Code_natural )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_4789_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_4790_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_4791_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_4792_mod__mult__eq,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C2 ) @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_4793_mod__mult__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C2 ) @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_4794_mod__mult__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C2 ) @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 ) ) ).

% mod_mult_eq
thf(fact_4795_mod__mult__cong,axiom,
    ! [A: nat,C2: nat,A2: nat,B: nat,B2: nat] :
      ( ( ( modulo_modulo_nat @ A @ C2 )
        = ( modulo_modulo_nat @ A2 @ C2 ) )
     => ( ( ( modulo_modulo_nat @ B @ C2 )
          = ( modulo_modulo_nat @ B2 @ C2 ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C2 )
          = ( modulo_modulo_nat @ ( times_times_nat @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_4796_mod__mult__cong,axiom,
    ! [A: int,C2: int,A2: int,B: int,B2: int] :
      ( ( ( modulo_modulo_int @ A @ C2 )
        = ( modulo_modulo_int @ A2 @ C2 ) )
     => ( ( ( modulo_modulo_int @ B @ C2 )
          = ( modulo_modulo_int @ B2 @ C2 ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C2 )
          = ( modulo_modulo_int @ ( times_times_int @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_4797_mod__mult__cong,axiom,
    ! [A: code_natural,C2: code_natural,A2: code_natural,B: code_natural,B2: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ C2 )
        = ( modulo8411746178871703098atural @ A2 @ C2 ) )
     => ( ( ( modulo8411746178871703098atural @ B @ C2 )
          = ( modulo8411746178871703098atural @ B2 @ C2 ) )
       => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 )
          = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A2 @ B2 ) @ C2 ) ) ) ) ).

% mod_mult_cong
thf(fact_4798_mod__mult__mult2,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ C2 ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_4799_mod__mult__mult2,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_4800_mod__mult__mult2,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ C2 ) @ ( times_2397367101498566445atural @ B @ C2 ) )
      = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ B ) @ C2 ) ) ).

% mod_mult_mult2
thf(fact_4801_mult__mod__right,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C2 @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C2 @ A ) @ ( times_times_nat @ C2 @ B ) ) ) ).

% mult_mod_right
thf(fact_4802_mult__mod__right,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( times_times_int @ C2 @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) ) ) ).

% mult_mod_right
thf(fact_4803_mult__mod__right,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( ( times_2397367101498566445atural @ C2 @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ C2 @ A ) @ ( times_2397367101498566445atural @ C2 @ B ) ) ) ).

% mult_mod_right
thf(fact_4804_mod__mult__left__eq,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C2 ) @ B ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_4805_mod__mult__left__eq,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C2 ) @ B ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_4806_mod__mult__left__eq,axiom,
    ! [A: code_natural,C2: code_natural,B: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ C2 ) @ B ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 ) ) ).

% mod_mult_left_eq
thf(fact_4807_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_4808_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_4809_mod__mult__right__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 )
      = ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 ) ) ).

% mod_mult_right_eq
thf(fact_4810_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_4811_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_4812_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_4813_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_4814_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_4815_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4816_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_4817_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4818_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4819_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = A )
      = ( ( divide5121882707175180666atural @ A @ B )
        = zero_z2226904508553997617atural ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_4820_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4821_cong__exp__iff__simps_I9_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_4822_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_4823_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_4824_mod__eqE,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C2 )
        = ( modulo_modulo_int @ B @ C2 ) )
     => ~ ! [D5: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C2 @ D5 ) ) ) ) ).

% mod_eqE
thf(fact_4825_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_4826_mod__induct,axiom,
    ! [P2: nat > $o,N: nat,P: nat,M2: nat] :
      ( ( P2 @ N )
     => ( ( ord_less_nat @ N @ P )
       => ( ( ord_less_nat @ M2 @ P )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P ) ) ) )
           => ( P2 @ M2 ) ) ) ) ) ).

% mod_induct
thf(fact_4827_mod__less__divisor,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M2 @ N ) @ N ) ) ).

% mod_less_divisor
thf(fact_4828_gcd__nat__induct,axiom,
    ! [P2: nat > nat > $o,M2: nat,N: nat] :
      ( ! [M: nat] : ( P2 @ M @ zero_zero_nat )
     => ( ! [M: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P2 @ N3 @ ( modulo_modulo_nat @ M @ N3 ) )
             => ( P2 @ M @ N3 ) ) )
       => ( P2 @ M2 @ N ) ) ) ).

% gcd_nat_induct
thf(fact_4829_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_4830_mod__eq__0D,axiom,
    ! [M2: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ D )
        = zero_zero_nat )
     => ? [Q3: nat] :
          ( M2
          = ( times_times_nat @ D @ Q3 ) ) ) ).

% mod_eq_0D
thf(fact_4831_mod__geq,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( modulo_modulo_nat @ M2 @ N )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ).

% mod_geq
thf(fact_4832_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_4833_nat__mod__eq__iff,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X @ ( times_times_nat @ N @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_4834_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_4835_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_4836_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_4837_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_4838_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4839_cong__exp__iff__simps_I2_J,axiom,
    ! [N: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_4840_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_4841_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_4842_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4843_cong__exp__iff__simps_I6_J,axiom,
    ! [Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_4844_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4845_cong__exp__iff__simps_I8_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_4846_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_4847_div__mult1__eq,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_4848_div__mult1__eq,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ B ) @ C2 )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ A @ ( divide5121882707175180666atural @ B @ C2 ) ) @ ( divide5121882707175180666atural @ ( times_2397367101498566445atural @ A @ ( modulo8411746178871703098atural @ B @ C2 ) ) @ C2 ) ) ) ).

% div_mult1_eq
thf(fact_4849_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
      = ( plus_plus_nat @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4850_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
      = ( plus_plus_int @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4851_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_natural,A: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A @ C2 ) ) ).

% cancel_div_mod_rules(2)
thf(fact_4852_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C2 )
      = ( plus_plus_nat @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4853_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C2 )
      = ( plus_plus_int @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4854_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_natural,B: code_natural,C2: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) @ C2 )
      = ( plus_p4538020629002901425atural @ A @ C2 ) ) ).

% cancel_div_mod_rules(1)
thf(fact_4855_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_4856_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_4857_mod__div__decomp,axiom,
    ! [A: code_natural,B: code_natural] :
      ( A
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_4858_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_4859_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_4860_div__mult__mod__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_4861_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_4862_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_4863_mod__div__mult__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_4864_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_4865_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_4866_mod__mult__div__eq,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ B ) @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_4867_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_4868_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_4869_mult__div__mod__eq,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) @ ( modulo8411746178871703098atural @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_4870_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4871_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4872_minus__div__mult__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4873_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4874_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4875_minus__mod__eq__div__mult,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4876_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4877_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4878_minus__mod__eq__mult__div,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) )
      = ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4879_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4880_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4881_minus__mult__div__eq__mod,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( minus_7197305767214868737atural @ A @ ( times_2397367101498566445atural @ B @ ( divide5121882707175180666atural @ A @ B ) ) )
      = ( modulo8411746178871703098atural @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4882_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_4883_div__less__mono,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( modulo_modulo_nat @ A4 @ N )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B5 @ N )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A4 @ N ) @ ( divide_divide_nat @ B5 @ N ) ) ) ) ) ) ).

% div_less_mono
thf(fact_4884_nat__mod__eq__lemma,axiom,
    ! [X: nat,N: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X @ N )
        = ( modulo_modulo_nat @ Y @ N ) )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ? [Q3: nat] :
            ( X
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N @ Q3 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_4885_mod__eq__nat2E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ~ ! [S: nat] :
              ( N
             != ( plus_plus_nat @ M2 @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_4886_mod__eq__nat1E,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( ( modulo_modulo_nat @ M2 @ Q2 )
        = ( modulo_modulo_nat @ N @ Q2 ) )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ~ ! [S: nat] :
              ( M2
             != ( plus_plus_nat @ N @ ( times_times_nat @ Q2 @ S ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_4887_mod__mult2__eq,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( modulo_modulo_nat @ M2 @ ( times_times_nat @ N @ Q2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N @ ( modulo_modulo_nat @ ( divide_divide_nat @ M2 @ N ) @ Q2 ) ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ).

% mod_mult2_eq
thf(fact_4888_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_4889_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N4: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N4 ) @ N4 ) ) ) ) ).

% modulo_nat_def
thf(fact_4890_mod__mult2__eq_H,axiom,
    ! [A: code_natural,M2: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ A @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( semiri3763490453095760265atural @ N ) ) )
      = ( plus_p4538020629002901425atural @ ( times_2397367101498566445atural @ ( semiri3763490453095760265atural @ M2 ) @ ( modulo8411746178871703098atural @ ( divide5121882707175180666atural @ A @ ( semiri3763490453095760265atural @ M2 ) ) @ ( semiri3763490453095760265atural @ N ) ) ) @ ( modulo8411746178871703098atural @ A @ ( semiri3763490453095760265atural @ M2 ) ) ) ) ).

% mod_mult2_eq'
thf(fact_4891_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_4892_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_4893_split__mod,axiom,
    ! [P2: nat > $o,M2: nat,N: nat] :
      ( ( P2 @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ M2 ) )
        & ( ( N != zero_zero_nat )
         => ! [I: nat,J: nat] :
              ( ( ord_less_nat @ J @ N )
             => ( ( M2
                  = ( plus_plus_nat @ ( times_times_nat @ N @ I ) @ J ) )
               => ( P2 @ J ) ) ) ) ) ) ).

% split_mod
thf(fact_4894_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4895_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4896_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_4897_nth__rotate1,axiom,
    ! [N: nat,Xs2: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( nth_VEBT_VEBT @ ( rotate1_VEBT_VEBT @ Xs2 ) @ N )
        = ( nth_VEBT_VEBT @ Xs2 @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ).

% nth_rotate1
thf(fact_4898_nth__rotate1,axiom,
    ! [N: nat,Xs2: list_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
     => ( ( nth_o @ ( rotate1_o @ Xs2 ) @ N )
        = ( nth_o @ Xs2 @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_o @ Xs2 ) ) ) ) ) ).

% nth_rotate1
thf(fact_4899_nth__rotate1,axiom,
    ! [N: nat,Xs2: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
     => ( ( nth_nat @ ( rotate1_nat @ Xs2 ) @ N )
        = ( nth_nat @ Xs2 @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ).

% nth_rotate1
thf(fact_4900_nth__rotate1,axiom,
    ! [N: nat,Xs2: list_int] :
      ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
     => ( ( nth_int @ ( rotate1_int @ Xs2 ) @ N )
        = ( nth_int @ Xs2 @ ( modulo_modulo_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs2 ) ) ) ) ) ).

% nth_rotate1
thf(fact_4901_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_4902_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_4903_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_4904_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_4905_bits__stable__imp__add__self,axiom,
    ! [A: code_natural] :
      ( ( ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p4538020629002901425atural @ A @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) )
        = zero_z2226904508553997617atural ) ) ).

% bits_stable_imp_add_self
thf(fact_4906_verit__le__mono__div,axiom,
    ! [A4: nat,B5: nat,N: nat] :
      ( ( ord_less_nat @ A4 @ B5 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A4 @ N )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B5 @ N )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B5 @ N ) ) ) ) ).

% verit_le_mono_div
thf(fact_4907_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_4908_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_4909_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_4910_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_4911_mult__exp__mod__exp__eq,axiom,
    ! [M2: nat,N: nat,A: code_natural] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo8411746178871703098atural @ ( times_2397367101498566445atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
        = ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4912_mod__double__modulus,axiom,
    ! [M2: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X )
       => ( ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_nat @ X @ M2 ) )
          | ( ( modulo_modulo_nat @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4913_mod__double__modulus,axiom,
    ! [M2: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ M2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ X )
       => ( ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( modulo_modulo_int @ X @ M2 ) )
          | ( ( modulo_modulo_int @ X @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M2 ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X @ M2 ) @ M2 ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4914_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_4915_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_4916_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_4917_unset__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se7083795435491715335atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se7083795435491715335atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4918_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_4919_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_4920_set__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se1617098188084679374atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se1617098188084679374atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4921_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_4922_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_4923_card__nth__roots,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z4: complex] :
                  ( ( power_power_complex @ Z4 @ N )
                  = C2 ) ) )
          = N ) ) ) ).

% card_nth_roots
thf(fact_4924_product__nth,axiom,
    ! [N: nat,Xs2: list_Code_integer,Ys: list_Code_integer] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs2 ) @ ( size_s3445333598471063425nteger @ Ys ) ) )
     => ( ( nth_Pr2304437835452373666nteger @ ( produc8792966785426426881nteger @ Xs2 @ Ys ) @ N )
        = ( produc1086072967326762835nteger @ ( nth_Code_integer @ Xs2 @ ( divide_divide_nat @ N @ ( size_s3445333598471063425nteger @ Ys ) ) ) @ ( nth_Code_integer @ Ys @ ( modulo_modulo_nat @ N @ ( size_s3445333598471063425nteger @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4925_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_4926_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_4927_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_4928_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_4929_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_4930_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_4931_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_4932_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_4933_product__nth,axiom,
    ! [N: nat,Xs2: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N @ ( times_times_nat @ ( size_size_list_nat @ Xs2 ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs2 @ Ys ) @ N )
        = ( produc599794634098209291T_VEBT @ ( nth_nat @ 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_4934_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_4935_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_4936_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_4937_VEBT__internal_Oinsert_H_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_insert @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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 ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) ) )
                    & ( ~ ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) @ Xa2 )
                     => ( Y
                        = ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_insert_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.insert'.pelims
thf(fact_4938_vebt__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_vebt_member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) @ Xa2 ) ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ( ~ Y
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ( ( Y
                          = ( ( Xa2 != Mi2 )
                           => ( ( Xa2 != Ma2 )
                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                         => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_4939_vebt__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw ) @ Xa2 ) ) )
           => ( ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz2 ) @ Xa2 ) ) )
             => ( ! [V: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
                       => ( ( Xa2 != Mi2 )
                         => ( ( Xa2 != Ma2 )
                           => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                              & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                               => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                  & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_4940_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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 ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) @ Xa2 ) ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_4941_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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 ) ) ) ) ) )
         => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ Xa2 ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_4942_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) )
               => ( ( ( Xa2 = zero_zero_nat )
                   => A3 )
                  & ( ( Xa2 != zero_zero_nat )
                   => ( ( ( Xa2 = one_one_nat )
                       => B3 )
                      & ( Xa2 = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw ) @ Xa2 ) ) )
           => ~ ! [Uy: option4927543243414619207at_nat,V: nat,TreeList: list_VEBT_VEBT,S: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ Xa2 ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_4943_vebt__member_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( 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,Va: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Xa2 != Mi2 )
                     => ( ( Xa2 != Ma2 )
                       => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                          & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
                           => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                              & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
                               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                                   => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_4944_zmod__numeral__Bit0,axiom,
    ! [V2: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_4945_zmod__le__nonneg__dividend,axiom,
    ! [M2: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M2 )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M2 @ K2 ) @ M2 ) ) ).

% zmod_le_nonneg_dividend
thf(fact_4946_zmod__eq__0__iff,axiom,
    ! [M2: int,D: int] :
      ( ( ( modulo_modulo_int @ M2 @ D )
        = zero_zero_int )
      = ( ? [Q5: int] :
            ( M2
            = ( times_times_int @ D @ Q5 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_4947_zmod__eq__0D,axiom,
    ! [M2: int,D: int] :
      ( ( ( modulo_modulo_int @ M2 @ D )
        = zero_zero_int )
     => ? [Q3: int] :
          ( M2
          = ( times_times_int @ D @ Q3 ) ) ) ).

% zmod_eq_0D
thf(fact_4948_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_4949_mod__int__unique,axiom,
    ! [K2: int,L: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( modulo_modulo_int @ K2 @ L )
        = R3 ) ) ).

% mod_int_unique
thf(fact_4950_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_4951_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_4952_zmod__trivial__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( ( modulo_modulo_int @ I2 @ K2 )
        = I2 )
      = ( ( K2 = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K2 ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K2 @ I2 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_4953_div__mod__decomp__int,axiom,
    ! [A4: int,N: int] :
      ( A4
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A4 @ N ) @ N ) @ ( modulo_modulo_int @ A4 @ N ) ) ) ).

% div_mod_decomp_int
thf(fact_4954_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_4955_mod__pos__geq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K2 )
       => ( ( modulo_modulo_int @ K2 @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K2 @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_4956_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_4957_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q2 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_4958_split__zmod,axiom,
    ! [P2: int > $o,N: int,K2: int] :
      ( ( P2 @ ( modulo_modulo_int @ N @ K2 ) )
      = ( ( ( K2 = zero_zero_int )
         => ( P2 @ N ) )
        & ( ( ord_less_int @ zero_zero_int @ K2 )
         => ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ J ) ) )
        & ( ( ord_less_int @ K2 @ zero_zero_int )
         => ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ J ) ) ) ) ) ).

% split_zmod
thf(fact_4959_zmod__zmult2__eq,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C2 )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_4960_split__pos__lemma,axiom,
    ! [K2: int,P2: int > int > $o,N: int] :
      ( ( ord_less_int @ zero_zero_int @ K2 )
     => ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
        = ( ! [I: int,J: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J )
                & ( ord_less_int @ J @ K2 )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I @ J ) ) ) ) ) ).

% split_pos_lemma
thf(fact_4961_split__neg__lemma,axiom,
    ! [K2: int,P2: int > int > $o,N: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ( ( P2 @ ( divide_divide_int @ N @ K2 ) @ ( modulo_modulo_int @ N @ K2 ) )
        = ( ! [I: int,J: int] :
              ( ( ( ord_less_int @ K2 @ J )
                & ( ord_less_eq_int @ J @ zero_zero_int )
                & ( N
                  = ( plus_plus_int @ ( times_times_int @ K2 @ I ) @ J ) ) )
             => ( P2 @ I @ J ) ) ) ) ) ).

% split_neg_lemma
thf(fact_4962_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_4963_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_4964_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa2 ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( 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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) )
                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) @ Xa2 ) ) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) )
                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_4965_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Xa2 ) ) )
           => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X
                    = ( 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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( ( X
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) )
               => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ Xa2 ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_4966_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va2: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X
                = ( 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,TreeList: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ Vc2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V ) @ TreeList @ 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 @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) )
           => ~ ! [V: nat,TreeList: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList @ Vd ) @ Xa2 ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList @ ( 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 @ TreeList ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_4967_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_4968_lemma__termdiff3,axiom,
    ! [H: real,Z3: real,K5: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z3 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z3 @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H ) @ N ) @ ( power_power_real @ Z3 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z3 @ ( 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_4969_lemma__termdiff3,axiom,
    ! [H: complex,Z3: complex,K5: real,N: nat] :
      ( ( H != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z3 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z3 @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H ) @ N ) @ ( power_power_complex @ Z3 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z3 @ ( 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_4970_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_4971_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_4972_flip__bit__Suc,axiom,
    ! [N: nat,A: code_natural] :
      ( ( bit_se168947363167071951atural @ ( suc @ N ) @ A )
      = ( plus_p4538020629002901425atural @ ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) @ ( times_2397367101498566445atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ ( bit_se168947363167071951atural @ N @ ( divide5121882707175180666atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4973_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_4974_norm__mult__numeral1,axiom,
    ! [W2: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_4975_norm__mult__numeral1,axiom,
    ! [W2: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_4976_norm__mult__numeral2,axiom,
    ! [A: real,W2: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).

% norm_mult_numeral2
thf(fact_4977_norm__mult__numeral2,axiom,
    ! [A: complex,W2: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W2 ) ) ) ).

% norm_mult_numeral2
thf(fact_4978_norm__le__zero__iff,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X ) @ zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_4979_norm__le__zero__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X ) @ zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_4980_zero__less__norm__iff,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X ) )
      = ( X != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_4981_zero__less__norm__iff,axiom,
    ! [X: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X ) )
      = ( X != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_4982_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4983_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_4984_norm__eq__zero,axiom,
    ! [X: real] :
      ( ( ( real_V7735802525324610683m_real @ X )
        = zero_zero_real )
      = ( X = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_4985_norm__eq__zero,axiom,
    ! [X: complex] :
      ( ( ( real_V1022390504157884413omplex @ X )
        = zero_zero_real )
      = ( X = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_4986_norm__power__diff,axiom,
    ! [Z3: real,W2: real,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W2 ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z3 @ M2 ) @ ( power_power_real @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z3 @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_4987_norm__power__diff,axiom,
    ! [Z3: complex,W2: complex,M2: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W2 ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z3 @ M2 ) @ ( power_power_complex @ W2 @ M2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z3 @ W2 ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_4988_norm__mult,axiom,
    ! [X: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult
thf(fact_4989_norm__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult
thf(fact_4990_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4991_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_4992_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N: nat,Z3: real] :
      ( ( ( power_power_real @ W2 @ N )
        = ( power_power_real @ Z3 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z3 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4993_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N: nat,Z3: complex] :
      ( ( ( power_power_complex @ W2 @ N )
        = ( power_power_complex @ Z3 @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z3 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_4994_norm__mult__less,axiom,
    ! [X: real,R3: real,Y: real,S2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ R3 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_4995_norm__mult__less,axiom,
    ! [X: complex,R3: real,Y: complex,S2: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S2 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ R3 @ S2 ) ) ) ) ).

% norm_mult_less
thf(fact_4996_norm__mult__ineq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_4997_norm__mult__ineq,axiom,
    ! [X: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_4998_power__eq__1__iff,axiom,
    ! [W2: real,N: nat] :
      ( ( ( power_power_real @ W2 @ N )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W2 )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_4999_power__eq__1__iff,axiom,
    ! [W2: complex,N: nat] :
      ( ( ( power_power_complex @ W2 @ N )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W2 )
          = one_one_real )
        | ( N = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5000_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_5001_vebt__buildup_Oelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va: nat] :
                ( ( X
                  = ( suc @ ( suc @ Va ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                    & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                     => ( Y
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.elims
thf(fact_5002_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > code_integer,Y: real > code_integer] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5003_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > code_integer,Y: nat > code_integer] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5004_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > code_integer,Y: int > code_integer] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5005_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > code_integer,Y: complex > code_integer] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I5 )
              & ( ( X @ I )
               != one_one_Code_integer ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_Code_integer ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( times_3573771949741848930nteger @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_Code_integer ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5006_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5007_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5008_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5009_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I5 )
              & ( ( X @ I )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( times_times_complex @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5010_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5011_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( times_times_real @ ( X @ I ) @ ( Y @ I ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_5012_sum__gp,axiom,
    ! [N: nat,M2: nat,X: complex] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5013_sum__gp,axiom,
    ! [N: nat,M2: nat,X: rat] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5014_sum__gp,axiom,
    ! [N: nat,M2: nat,X: real] :
      ( ( ( ord_less_nat @ N @ M2 )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N @ M2 )
       => ( ( ( X = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ M2 ) ) ) )
          & ( ( X != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_5015_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > complex,Y: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5016_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > complex,Y: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5017_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > complex,Y: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5018_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > complex,Y: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( plus_plus_complex @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5019_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > real,Y: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5020_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > real,Y: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5021_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X: int > real,Y: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I: int] :
              ( ( member_int @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I: int] :
                ( ( member_int @ I @ I5 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5022_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X: complex > real,Y: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I: complex] :
              ( ( member_complex @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I: complex] :
                ( ( member_complex @ I @ I5 )
                & ( ( plus_plus_real @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5023_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X: real > rat,Y: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I: real] :
              ( ( member_real @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I: real] :
                ( ( member_real @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5024_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X: nat > rat,Y: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ I5 )
              & ( ( X @ I )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( Y @ I )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I: nat] :
                ( ( member_nat @ I @ I5 )
                & ( ( plus_plus_rat @ ( X @ I ) @ ( Y @ I ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_5025_geometric__deriv__sums,axiom,
    ! [Z3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z3 ) @ one_one_real )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) @ ( power_power_real @ Z3 @ N4 ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5026_geometric__deriv__sums,axiom,
    ! [Z3: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z3 ) @ one_one_real )
     => ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N4 ) ) @ ( power_power_complex @ Z3 @ N4 ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_5027_pochhammer__double,axiom,
    ! [Z3: complex,N: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z3 ) @ ( 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 @ Z3 @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5028_pochhammer__double,axiom,
    ! [Z3: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z3 ) @ ( 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 @ Z3 @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5029_pochhammer__double,axiom,
    ! [Z3: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z3 ) @ ( 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 @ Z3 @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N ) ) ) ).

% pochhammer_double
thf(fact_5030_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_5031_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_5032_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_5033_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_5034_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_5035_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_5036_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_5037_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_5038_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_5039_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_5040_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_5041_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_5042_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_5043_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_5044_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_5045_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_5046_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_5047_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_5048_div__dvd__div,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C2 )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C2 @ A ) )
          = ( dvd_dvd_nat @ B @ C2 ) ) ) ) ).

% div_dvd_div
thf(fact_5049_div__dvd__div,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C2 )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C2 @ A ) )
          = ( dvd_dvd_int @ B @ C2 ) ) ) ) ).

% div_dvd_div
thf(fact_5050_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_5051_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A4 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_5052_sum_Oneutral__const,axiom,
    ! [A4: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [Uu3: int] : zero_zero_int
        @ A4 )
      = zero_zero_int ) ).

% sum.neutral_const
thf(fact_5053_sum_Oneutral__const,axiom,
    ! [A4: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A4 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_5054_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C2 @ A ) )
        = ( dvd_dvd_nat @ B @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5055_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C2 @ A ) )
        = ( dvd_dvd_int @ B @ C2 ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_5056_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C2 ) )
        = ( dvd_dvd_nat @ B @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5057_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C2 ) )
        = ( dvd_dvd_int @ B @ C2 ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_5058_dvd__mult__cancel__right,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C2 ) @ ( times_times_complex @ B @ C2 ) )
      = ( ( C2 = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5059_dvd__mult__cancel__right,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ C2 ) )
      = ( ( C2 = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5060_dvd__mult__cancel__right,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ C2 ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5061_dvd__mult__cancel__right,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ C2 ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_5062_dvd__mult__cancel__left,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C2 @ A ) @ ( times_times_complex @ C2 @ B ) )
      = ( ( C2 = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5063_dvd__mult__cancel__left,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C2 @ A ) @ ( times_times_real @ C2 @ B ) )
      = ( ( C2 = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5064_dvd__mult__cancel__left,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C2 @ A ) @ ( times_times_rat @ C2 @ B ) )
      = ( ( C2 = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5065_dvd__mult__cancel__left,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
      = ( ( C2 = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_5066_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C2 @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5067_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C2 @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5068_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C2 @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5069_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C2 @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_5070_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C2 @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5071_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C2 @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5072_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C2 @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5073_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C2 @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_5074_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_5075_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_5076_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_5077_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_5078_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_5079_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_5080_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_5081_div__add,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ A )
     => ( ( dvd_dvd_nat @ C2 @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C2 ) @ ( divide_divide_nat @ B @ C2 ) ) ) ) ) ).

% div_add
thf(fact_5082_div__add,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ A )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C2 )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).

% div_add
thf(fact_5083_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_5084_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_5085_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_5086_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_5087_unit__div__1__unit,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).

% unit_div_1_unit
thf(fact_5088_unit__div__1__unit,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).

% unit_div_1_unit
thf(fact_5089_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_5090_unit__div__1__div__1,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_5091_unit__div__1__div__1,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_5092_sum_Oempty,axiom,
    ! [G2: nat > complex] :
      ( ( groups2073611262835488442omplex @ G2 @ bot_bot_set_nat )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_5093_sum_Oempty,axiom,
    ! [G2: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G2 @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_5094_sum_Oempty,axiom,
    ! [G2: nat > int] :
      ( ( groups3539618377306564664at_int @ G2 @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_5095_sum_Oempty,axiom,
    ! [G2: int > complex] :
      ( ( groups3049146728041665814omplex @ G2 @ bot_bot_set_int )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_5096_sum_Oempty,axiom,
    ! [G2: int > real] :
      ( ( groups8778361861064173332t_real @ G2 @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_5097_sum_Oempty,axiom,
    ! [G2: int > rat] :
      ( ( groups3906332499630173760nt_rat @ G2 @ bot_bot_set_int )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_5098_sum_Oempty,axiom,
    ! [G2: int > nat] :
      ( ( groups4541462559716669496nt_nat @ G2 @ bot_bot_set_int )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_5099_sum_Oempty,axiom,
    ! [G2: real > complex] :
      ( ( groups5754745047067104278omplex @ G2 @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_5100_sum_Oempty,axiom,
    ! [G2: real > real] :
      ( ( groups8097168146408367636l_real @ G2 @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_5101_sum_Oempty,axiom,
    ! [G2: real > rat] :
      ( ( groups1300246762558778688al_rat @ G2 @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_5102_sum_Oinfinite,axiom,
    ! [A4: set_nat,G2: nat > complex] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups2073611262835488442omplex @ G2 @ A4 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_5103_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: int > complex] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups3049146728041665814omplex @ G2 @ A4 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_5104_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex @ G2 @ A4 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_5105_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: int > real] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G2 @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_5106_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G2 @ A4 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_5107_sum_Oinfinite,axiom,
    ! [A4: set_nat,G2: nat > rat] :
      ( ~ ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G2 @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_5108_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: int > rat] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G2 @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_5109_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G2 @ A4 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_5110_sum_Oinfinite,axiom,
    ! [A4: set_int,G2: int > nat] :
      ( ~ ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat @ G2 @ A4 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_5111_sum_Oinfinite,axiom,
    ! [A4: set_complex,G2: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat @ G2 @ A4 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_5112_sum__eq__0__iff,axiom,
    ! [F3: set_int,F2: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F2 @ F3 )
          = zero_zero_nat )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ F3 )
             => ( ( F2 @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5113_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F2 @ F3 )
          = zero_zero_nat )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ F3 )
             => ( ( F2 @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5114_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F2 @ F3 )
          = zero_zero_nat )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ F3 )
             => ( ( F2 @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_5115_div__diff,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ A )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C2 )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C2 ) @ ( divide_divide_int @ B @ C2 ) ) ) ) ) ).

% div_diff
thf(fact_5116_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_5117_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_5118_dvd__imp__mod__0,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ A @ B )
     => ( ( modulo8411746178871703098atural @ B @ A )
        = zero_z2226904508553997617atural ) ) ).

% dvd_imp_mod_0
thf(fact_5119_dvd__1__left,axiom,
    ! [K2: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K2 ) ).

% dvd_1_left
thf(fact_5120_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_5121_nat__mult__dvd__cancel__disj,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
      = ( ( K2 = zero_zero_nat )
        | ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_5122_pochhammer__0,axiom,
    ! [A: code_integer] :
      ( ( comm_s8582702949713902594nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% pochhammer_0
thf(fact_5123_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_5124_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_5125_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5126_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5127_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_5128_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_5129_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_5130_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups7754918857620584856omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups7754918857620584856omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_5131_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5132_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5133_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_5134_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_5135_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_5136_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_5137_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_5138_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_5139_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_5140_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups7754918857620584856omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups7754918857620584856omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_5141_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5142_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5143_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_5144_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_5145_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_5146_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_5147_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_5148_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_5149_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_5150_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_5151_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_5152_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_5153_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_5154_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_5155_sum__constant,axiom,
    ! [Y: rat,A4: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5156_sum__constant,axiom,
    ! [Y: rat,A4: set_complex] :
      ( ( groups5058264527183730370ex_rat
        @ ^ [X4: complex] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5157_sum__constant,axiom,
    ! [Y: rat,A4: set_int] :
      ( ( groups3906332499630173760nt_rat
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5158_sum__constant,axiom,
    ! [Y: real,A4: set_complex] :
      ( ( groups5808333547571424918x_real
        @ ^ [X4: complex] : Y
        @ A4 )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5159_sum__constant,axiom,
    ! [Y: real,A4: set_int] :
      ( ( groups8778361861064173332t_real
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5160_sum__constant,axiom,
    ! [Y: int,A4: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5161_sum__constant,axiom,
    ! [Y: int,A4: set_complex] :
      ( ( groups5690904116761175830ex_int
        @ ^ [X4: complex] : Y
        @ A4 )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5162_sum__constant,axiom,
    ! [Y: nat,A4: set_complex] :
      ( ( groups5693394587270226106ex_nat
        @ ^ [X4: complex] : Y
        @ A4 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5163_sum__constant,axiom,
    ! [Y: nat,A4: set_int] :
      ( ( groups4541462559716669496nt_nat
        @ ^ [X4: int] : Y
        @ A4 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5164_sum__constant,axiom,
    ! [Y: nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : Y
        @ A4 )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_nat @ A4 ) ) @ Y ) ) ).

% sum_constant
thf(fact_5165_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_5166_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_5167_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_5168_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_5169_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_5170_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
        @ X )
      = ( ( A @ zero_zero_nat )
        = X ) ) ).

% powser_sums_zero_iff
thf(fact_5171_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_5172_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_5173_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2073611262835488442omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5174_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5175_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5176_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5177_sum_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_5178_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_5179_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_5180_sum__zero__power,axiom,
    ! [A4: set_nat,C2: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ zero_zero_real @ I ) )
            @ A4 )
          = ( C2 @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ zero_zero_real @ I ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_5181_zero__le__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5182_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5183_zero__le__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_5184_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_5185_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_5186_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_5187_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_5188_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_5189_power__less__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5190_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5191_power__less__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_5192_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_5193_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: nat > complex,D: nat > complex] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) ) @ ( D @ I ) )
            @ A4 )
          = ( divide1717551699836669952omplex @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ zero_zero_complex @ I ) ) @ ( D @ I ) )
            @ A4 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_5194_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D @ I ) )
            @ A4 )
          = ( divide_divide_rat @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ zero_zero_rat @ I ) ) @ ( D @ I ) )
            @ A4 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_5195_sum__zero__power_H,axiom,
    ! [A4: set_nat,C2: nat > real,D: nat > real] :
      ( ( ( ( finite_finite_nat @ A4 )
          & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ zero_zero_real @ I ) ) @ ( D @ I ) )
            @ A4 )
          = ( divide_divide_real @ ( C2 @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A4 )
            & ( member_nat @ zero_zero_nat @ A4 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( divide_divide_real @ ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ zero_zero_real @ I ) ) @ ( D @ I ) )
            @ A4 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_5196_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_5197_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_5198_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_5199_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_5200_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_5201_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_5202_zero__less__power__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5203_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5204_zero__less__power__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_5205_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_5206_power__le__zero__eq__numeral,axiom,
    ! [A: real,W2: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5207_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5208_power__le__zero__eq__numeral,axiom,
    ! [A: int,W2: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_5209_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_5210_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_5211_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_5212_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_5213_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_5214_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_5215_even__succ__mod__exp,axiom,
    ! [A: code_natural,N: nat] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( modulo8411746178871703098atural @ ( plus_p4538020629002901425atural @ one_one_Code_natural @ A ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
          = ( plus_p4538020629002901425atural @ one_one_Code_natural @ ( modulo8411746178871703098atural @ A @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_5216_sum_Oneutral,axiom,
    ! [A4: set_nat,G2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G2 @ X3 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G2 @ A4 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_5217_sum_Oneutral,axiom,
    ! [A4: set_int,G2: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( G2 @ X3 )
            = zero_zero_int ) )
     => ( ( groups4538972089207619220nt_int @ G2 @ A4 )
        = zero_zero_int ) ) ).

% sum.neutral
thf(fact_5218_sum_Oneutral,axiom,
    ! [A4: set_nat,G2: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( G2 @ X3 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G2 @ A4 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_5219_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: complex > complex,A4: set_complex] :
      ( ( ( groups7754918857620584856omplex @ G2 @ A4 )
       != zero_zero_complex )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5220_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: real > complex,A4: set_real] :
      ( ( ( groups5754745047067104278omplex @ G2 @ A4 )
       != zero_zero_complex )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5221_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > complex,A4: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G2 @ A4 )
       != zero_zero_complex )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5222_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > complex,A4: set_int] :
      ( ( ( groups3049146728041665814omplex @ G2 @ A4 )
       != zero_zero_complex )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5223_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: complex > real,A4: set_complex] :
      ( ( ( groups5808333547571424918x_real @ G2 @ A4 )
       != zero_zero_real )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5224_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: real > real,A4: set_real] :
      ( ( ( groups8097168146408367636l_real @ G2 @ A4 )
       != zero_zero_real )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5225_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: int > real,A4: set_int] :
      ( ( ( groups8778361861064173332t_real @ G2 @ A4 )
       != zero_zero_real )
     => ~ ! [A3: int] :
            ( ( member_int @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_real ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5226_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: complex > rat,A4: set_complex] :
      ( ( ( groups5058264527183730370ex_rat @ G2 @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: complex] :
            ( ( member_complex @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5227_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: real > rat,A4: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G2 @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: real] :
            ( ( member_real @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5228_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G2: nat > rat,A4: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G2 @ A4 )
       != zero_zero_rat )
     => ~ ! [A3: nat] :
            ( ( member_nat @ A3 @ A4 )
           => ( ( G2 @ A3 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_5229_sum__mono,axiom,
    ! [K5: set_complex,F2: complex > rat,G2: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F2 @ K5 ) @ ( groups5058264527183730370ex_rat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5230_sum__mono,axiom,
    ! [K5: set_real,F2: real > rat,G2: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F2 @ K5 ) @ ( groups1300246762558778688al_rat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5231_sum__mono,axiom,
    ! [K5: set_nat,F2: nat > rat,G2: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ K5 ) @ ( groups2906978787729119204at_rat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5232_sum__mono,axiom,
    ! [K5: set_int,F2: int > rat,G2: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ K5 ) @ ( groups3906332499630173760nt_rat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5233_sum__mono,axiom,
    ! [K5: set_complex,F2: complex > nat,G2: complex > nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F2 @ K5 ) @ ( groups5693394587270226106ex_nat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5234_sum__mono,axiom,
    ! [K5: set_real,F2: real > nat,G2: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F2 @ K5 ) @ ( groups1935376822645274424al_nat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5235_sum__mono,axiom,
    ! [K5: set_int,F2: int > nat,G2: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ K5 ) @ ( groups4541462559716669496nt_nat @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5236_sum__mono,axiom,
    ! [K5: set_complex,F2: complex > int,G2: complex > int] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F2 @ K5 ) @ ( groups5690904116761175830ex_int @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5237_sum__mono,axiom,
    ! [K5: set_real,F2: real > int,G2: real > int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F2 @ K5 ) @ ( groups1932886352136224148al_int @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5238_sum__mono,axiom,
    ! [K5: set_nat,F2: nat > int,G2: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ K5 ) @ ( groups3539618377306564664at_int @ G2 @ K5 ) ) ) ).

% sum_mono
thf(fact_5239_sum__distrib__left,axiom,
    ! [R3: nat,F2: nat > nat,A4: set_nat] :
      ( ( times_times_nat @ R3 @ ( groups3542108847815614940at_nat @ F2 @ A4 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ R3 @ ( F2 @ N4 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_5240_sum__distrib__left,axiom,
    ! [R3: int,F2: int > int,A4: set_int] :
      ( ( times_times_int @ R3 @ ( groups4538972089207619220nt_int @ F2 @ A4 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ R3 @ ( F2 @ N4 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_5241_sum__distrib__left,axiom,
    ! [R3: real,F2: nat > real,A4: set_nat] :
      ( ( times_times_real @ R3 @ ( groups6591440286371151544t_real @ F2 @ A4 ) )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ R3 @ ( F2 @ N4 ) )
        @ A4 ) ) ).

% sum_distrib_left
thf(fact_5242_sum__distrib__right,axiom,
    ! [F2: nat > nat,A4: set_nat,R3: nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F2 @ A4 ) @ R3 )
      = ( groups3542108847815614940at_nat
        @ ^ [N4: nat] : ( times_times_nat @ ( F2 @ N4 ) @ R3 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_5243_sum__distrib__right,axiom,
    ! [F2: int > int,A4: set_int,R3: int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F2 @ A4 ) @ R3 )
      = ( groups4538972089207619220nt_int
        @ ^ [N4: int] : ( times_times_int @ ( F2 @ N4 ) @ R3 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_5244_sum__distrib__right,axiom,
    ! [F2: nat > real,A4: set_nat,R3: real] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F2 @ A4 ) @ R3 )
      = ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( times_times_real @ ( F2 @ N4 ) @ R3 )
        @ A4 ) ) ).

% sum_distrib_right
thf(fact_5245_sum__product,axiom,
    ! [F2: nat > nat,A4: set_nat,G2: nat > nat,B5: set_nat] :
      ( ( times_times_nat @ ( groups3542108847815614940at_nat @ F2 @ A4 ) @ ( groups3542108847815614940at_nat @ G2 @ B5 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( times_times_nat @ ( F2 @ I ) @ ( G2 @ J ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_5246_sum__product,axiom,
    ! [F2: int > int,A4: set_int,G2: int > int,B5: set_int] :
      ( ( times_times_int @ ( groups4538972089207619220nt_int @ F2 @ A4 ) @ ( groups4538972089207619220nt_int @ G2 @ B5 ) )
      = ( groups4538972089207619220nt_int
        @ ^ [I: int] :
            ( groups4538972089207619220nt_int
            @ ^ [J: int] : ( times_times_int @ ( F2 @ I ) @ ( G2 @ J ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_5247_sum__product,axiom,
    ! [F2: nat > real,A4: set_nat,G2: nat > real,B5: set_nat] :
      ( ( times_times_real @ ( groups6591440286371151544t_real @ F2 @ A4 ) @ ( groups6591440286371151544t_real @ G2 @ B5 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] :
            ( groups6591440286371151544t_real
            @ ^ [J: nat] : ( times_times_real @ ( F2 @ I ) @ ( G2 @ J ) )
            @ B5 )
        @ A4 ) ) ).

% sum_product
thf(fact_5248_dvd__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ M2 @ N )
     => ( ( dvd_dvd_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% dvd_antisym
thf(fact_5249_dvd__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C2 )
       => ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_trans
thf(fact_5250_dvd__trans,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C2 )
       => ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_trans
thf(fact_5251_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_5252_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_5253_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A5: complex,B4: complex] :
          ( ( A5 = zero_zero_complex )
         => ( B4 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_5254_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A5: real,B4: real] :
          ( ( A5 = zero_zero_real )
         => ( B4 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_5255_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A5: rat,B4: rat] :
          ( ( A5 = zero_zero_rat )
         => ( B4 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_5256_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_5257_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_5258_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_5259_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_5260_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_5261_dvd__productE,axiom,
    ! [P: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P @ ( times_times_nat @ A @ B ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( times_times_nat @ X3 @ Y3 ) )
           => ( ( dvd_dvd_nat @ X3 @ A )
             => ~ ( dvd_dvd_nat @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5262_dvd__productE,axiom,
    ! [P: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P @ ( times_times_int @ A @ B ) )
     => ~ ! [X3: int,Y3: int] :
            ( ( P
              = ( times_times_int @ X3 @ Y3 ) )
           => ( ( dvd_dvd_int @ X3 @ A )
             => ~ ( dvd_dvd_int @ Y3 @ B ) ) ) ) ).

% dvd_productE
thf(fact_5263_division__decomp,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
     => ? [B9: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B9 @ C5 ) )
          & ( dvd_dvd_nat @ B9 @ B )
          & ( dvd_dvd_nat @ C5 @ C2 ) ) ) ).

% division_decomp
thf(fact_5264_division__decomp,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
     => ? [B9: int,C5: int] :
          ( ( A
            = ( times_times_int @ B9 @ C5 ) )
          & ( dvd_dvd_int @ B9 @ B )
          & ( dvd_dvd_int @ C5 @ C2 ) ) ) ).

% division_decomp
thf(fact_5265_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K: real] :
            ( A
           != ( times_times_real @ B @ K ) ) ) ).

% dvdE
thf(fact_5266_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K: rat] :
            ( A
           != ( times_times_rat @ B @ K ) ) ) ).

% dvdE
thf(fact_5267_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K: nat] :
            ( A
           != ( times_times_nat @ B @ K ) ) ) ).

% dvdE
thf(fact_5268_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K: int] :
            ( A
           != ( times_times_int @ B @ K ) ) ) ).

% dvdE
thf(fact_5269_dvdI,axiom,
    ! [A: real,B: real,K2: real] :
      ( ( A
        = ( times_times_real @ B @ K2 ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_5270_dvdI,axiom,
    ! [A: rat,B: rat,K2: rat] :
      ( ( A
        = ( times_times_rat @ B @ K2 ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_5271_dvdI,axiom,
    ! [A: nat,B: nat,K2: nat] :
      ( ( A
        = ( times_times_nat @ B @ K2 ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_5272_dvdI,axiom,
    ! [A: int,B: int,K2: int] :
      ( ( A
        = ( times_times_int @ B @ K2 ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_5273_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B4: real,A5: real] :
        ? [K3: real] :
          ( A5
          = ( times_times_real @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5274_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B4: rat,A5: rat] :
        ? [K3: rat] :
          ( A5
          = ( times_times_rat @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5275_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B4: nat,A5: nat] :
        ? [K3: nat] :
          ( A5
          = ( times_times_nat @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5276_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B4: int,A5: int] :
        ? [K3: int] :
          ( A5
          = ( times_times_int @ B4 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_5277_dvd__mult,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( dvd_dvd_real @ A @ C2 )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% dvd_mult
thf(fact_5278_dvd__mult,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C2 )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% dvd_mult
thf(fact_5279_dvd__mult,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C2 )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% dvd_mult
thf(fact_5280_dvd__mult,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ A @ C2 )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% dvd_mult
thf(fact_5281_dvd__mult2,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C2 ) ) ) ).

% dvd_mult2
thf(fact_5282_dvd__mult2,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C2 ) ) ) ).

% dvd_mult2
thf(fact_5283_dvd__mult2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) ) ) ).

% dvd_mult2
thf(fact_5284_dvd__mult2,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) ) ) ).

% dvd_mult2
thf(fact_5285_dvd__mult__left,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
     => ( dvd_dvd_real @ A @ C2 ) ) ).

% dvd_mult_left
thf(fact_5286_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C2 )
     => ( dvd_dvd_rat @ A @ C2 ) ) ).

% dvd_mult_left
thf(fact_5287_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
     => ( dvd_dvd_nat @ A @ C2 ) ) ).

% dvd_mult_left
thf(fact_5288_dvd__mult__left,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
     => ( dvd_dvd_int @ A @ C2 ) ) ).

% dvd_mult_left
thf(fact_5289_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5290_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5291_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5292_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_5293_mult__dvd__mono,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C2 @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5294_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C2 @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C2 ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5295_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C2 @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5296_mult__dvd__mono,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C2 @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_5297_dvd__mult__right,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C2 )
     => ( dvd_dvd_real @ B @ C2 ) ) ).

% dvd_mult_right
thf(fact_5298_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C2 )
     => ( dvd_dvd_rat @ B @ C2 ) ) ).

% dvd_mult_right
thf(fact_5299_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
     => ( dvd_dvd_nat @ B @ C2 ) ) ).

% dvd_mult_right
thf(fact_5300_dvd__mult__right,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
     => ( dvd_dvd_int @ B @ C2 ) ) ).

% dvd_mult_right
thf(fact_5301_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5302_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5303_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5304_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_5305_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
        = ( dvd_dvd_real @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_5306_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( dvd_dvd_rat @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_5307_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_5308_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_add_right_iff
thf(fact_5309_dvd__add__left__iff,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( dvd_dvd_real @ A @ C2 )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5310_dvd__add__left__iff,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C2 )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5311_dvd__add__left__iff,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C2 )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5312_dvd__add__left__iff,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ A @ C2 )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_5313_dvd__add,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C2 )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_5314_dvd__add,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C2 )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_5315_dvd__add,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C2 )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_5316_dvd__add,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C2 )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ) ).

% dvd_add
thf(fact_5317_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5318_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5319_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_5320_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_5321_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_5322_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_5323_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_5324_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_5325_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_5326_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_5327_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_5328_dvd__diff,axiom,
    ! [X: rat,Y: rat,Z3: rat] :
      ( ( dvd_dvd_rat @ X @ Y )
     => ( ( dvd_dvd_rat @ X @ Z3 )
       => ( dvd_dvd_rat @ X @ ( minus_minus_rat @ Y @ Z3 ) ) ) ) ).

% dvd_diff
thf(fact_5329_dvd__diff,axiom,
    ! [X: int,Y: int,Z3: int] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( dvd_dvd_int @ X @ Z3 )
       => ( dvd_dvd_int @ X @ ( minus_minus_int @ Y @ Z3 ) ) ) ) ).

% dvd_diff
thf(fact_5330_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_5331_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_5332_dvd__div__eq__cancel,axiom,
    ! [A: complex,C2: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C2 )
        = ( divide1717551699836669952omplex @ B @ C2 ) )
     => ( ( dvd_dvd_complex @ C2 @ A )
       => ( ( dvd_dvd_complex @ C2 @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5333_dvd__div__eq__cancel,axiom,
    ! [A: real,C2: real,B: real] :
      ( ( ( divide_divide_real @ A @ C2 )
        = ( divide_divide_real @ B @ C2 ) )
     => ( ( dvd_dvd_real @ C2 @ A )
       => ( ( dvd_dvd_real @ C2 @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5334_dvd__div__eq__cancel,axiom,
    ! [A: rat,C2: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C2 )
        = ( divide_divide_rat @ B @ C2 ) )
     => ( ( dvd_dvd_rat @ C2 @ A )
       => ( ( dvd_dvd_rat @ C2 @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5335_dvd__div__eq__cancel,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C2 )
        = ( divide_divide_nat @ B @ C2 ) )
     => ( ( dvd_dvd_nat @ C2 @ A )
       => ( ( dvd_dvd_nat @ C2 @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5336_dvd__div__eq__cancel,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( ( divide_divide_int @ A @ C2 )
        = ( divide_divide_int @ B @ C2 ) )
     => ( ( dvd_dvd_int @ C2 @ A )
       => ( ( dvd_dvd_int @ C2 @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_5337_dvd__div__eq__iff,axiom,
    ! [C2: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C2 @ A )
     => ( ( dvd_dvd_complex @ C2 @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C2 )
            = ( divide1717551699836669952omplex @ B @ C2 ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5338_dvd__div__eq__iff,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C2 @ A )
     => ( ( dvd_dvd_real @ C2 @ B )
       => ( ( ( divide_divide_real @ A @ C2 )
            = ( divide_divide_real @ B @ C2 ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5339_dvd__div__eq__iff,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C2 @ A )
     => ( ( dvd_dvd_rat @ C2 @ B )
       => ( ( ( divide_divide_rat @ A @ C2 )
            = ( divide_divide_rat @ B @ C2 ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5340_dvd__div__eq__iff,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ A )
     => ( ( dvd_dvd_nat @ C2 @ B )
       => ( ( ( divide_divide_nat @ A @ C2 )
            = ( divide_divide_nat @ B @ C2 ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5341_dvd__div__eq__iff,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ A )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( ( ( divide_divide_int @ A @ C2 )
            = ( divide_divide_int @ B @ C2 ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_5342_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_5343_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_5344_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_5345_gcd__nat_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
        & ( A != zero_zero_nat ) ) ) ).

% gcd_nat.not_eq_extremum
thf(fact_5346_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_5347_dvd__mod__imp__dvd,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C2 @ B )
       => ( dvd_dvd_nat @ C2 @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5348_dvd__mod__imp__dvd,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( dvd_dvd_int @ C2 @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5349_dvd__mod__imp__dvd,axiom,
    ! [C2: code_natural,A: code_natural,B: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ ( modulo8411746178871703098atural @ A @ B ) )
     => ( ( dvd_dvd_Code_natural @ C2 @ B )
       => ( dvd_dvd_Code_natural @ C2 @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_5350_dvd__mod__iff,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ B )
     => ( ( dvd_dvd_nat @ C2 @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C2 @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5351_dvd__mod__iff,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ B )
     => ( ( dvd_dvd_int @ C2 @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C2 @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5352_dvd__mod__iff,axiom,
    ! [C2: code_natural,B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ C2 @ B )
     => ( ( dvd_dvd_Code_natural @ C2 @ ( modulo8411746178871703098atural @ A @ B ) )
        = ( dvd_dvd_Code_natural @ C2 @ A ) ) ) ).

% dvd_mod_iff
thf(fact_5353_sum__nonneg,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5354_sum__nonneg,axiom,
    ! [A4: set_real,F2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5355_sum__nonneg,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5356_sum__nonneg,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5357_sum__nonneg,axiom,
    ! [A4: set_real,F2: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5358_sum__nonneg,axiom,
    ! [A4: set_nat,F2: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5359_sum__nonneg,axiom,
    ! [A4: set_int,F2: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5360_sum__nonneg,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5361_sum__nonneg,axiom,
    ! [A4: set_real,F2: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5362_sum__nonneg,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) ) ) ).

% sum_nonneg
thf(fact_5363_sum__nonpos,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5364_sum__nonpos,axiom,
    ! [A4: set_real,F2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5365_sum__nonpos,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ X3 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_5366_sum__nonpos,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_5367_sum__nonpos,axiom,
    ! [A4: set_real,F2: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_5368_sum__nonpos,axiom,
    ! [A4: set_nat,F2: nat > rat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_5369_sum__nonpos,axiom,
    ! [A4: set_int,F2: int > rat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ X3 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_5370_sum__nonpos,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5371_sum__nonpos,axiom,
    ! [A4: set_real,F2: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5372_sum__nonpos,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ X3 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_5373_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_5374_sum__mono__inv,axiom,
    ! [F2: real > rat,I5: set_real,G2: real > rat,I2: real] :
      ( ( ( groups1300246762558778688al_rat @ F2 @ I5 )
        = ( groups1300246762558778688al_rat @ G2 @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_real @ I2 @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5375_sum__mono__inv,axiom,
    ! [F2: nat > rat,I5: set_nat,G2: nat > rat,I2: nat] :
      ( ( ( groups2906978787729119204at_rat @ F2 @ I5 )
        = ( groups2906978787729119204at_rat @ G2 @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_nat @ I2 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5376_sum__mono__inv,axiom,
    ! [F2: int > rat,I5: set_int,G2: int > rat,I2: int] :
      ( ( ( groups3906332499630173760nt_rat @ F2 @ I5 )
        = ( groups3906332499630173760nt_rat @ G2 @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_int @ I2 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5377_sum__mono__inv,axiom,
    ! [F2: complex > rat,I5: set_complex,G2: complex > rat,I2: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F2 @ I5 )
        = ( groups5058264527183730370ex_rat @ G2 @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_complex @ I2 @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5378_sum__mono__inv,axiom,
    ! [F2: real > nat,I5: set_real,G2: real > nat,I2: real] :
      ( ( ( groups1935376822645274424al_nat @ F2 @ I5 )
        = ( groups1935376822645274424al_nat @ G2 @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_real @ I2 @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5379_sum__mono__inv,axiom,
    ! [F2: int > nat,I5: set_int,G2: int > nat,I2: int] :
      ( ( ( groups4541462559716669496nt_nat @ F2 @ I5 )
        = ( groups4541462559716669496nt_nat @ G2 @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_int @ I2 @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5380_sum__mono__inv,axiom,
    ! [F2: complex > nat,I5: set_complex,G2: complex > nat,I2: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F2 @ I5 )
        = ( groups5693394587270226106ex_nat @ G2 @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_complex @ I2 @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5381_sum__mono__inv,axiom,
    ! [F2: real > int,I5: set_real,G2: real > int,I2: real] :
      ( ( ( groups1932886352136224148al_int @ F2 @ I5 )
        = ( groups1932886352136224148al_int @ G2 @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_real @ I2 @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5382_sum__mono__inv,axiom,
    ! [F2: nat > int,I5: set_nat,G2: nat > int,I2: nat] :
      ( ( ( groups3539618377306564664at_int @ F2 @ I5 )
        = ( groups3539618377306564664at_int @ G2 @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_nat @ I2 @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5383_sum__mono__inv,axiom,
    ! [F2: complex > int,I5: set_complex,G2: complex > int,I2: complex] :
      ( ( ( groups5690904116761175830ex_int @ F2 @ I5 )
        = ( groups5690904116761175830ex_int @ G2 @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) )
       => ( ( member_complex @ I2 @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F2 @ I2 )
              = ( G2 @ I2 ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_5384_sum__cong__Suc,axiom,
    ! [A4: set_nat,F2: nat > nat,G2: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F2 @ ( suc @ X3 ) )
              = ( G2 @ ( suc @ X3 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F2 @ A4 )
          = ( groups3542108847815614940at_nat @ G2 @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5385_sum__cong__Suc,axiom,
    ! [A4: set_nat,F2: nat > real,G2: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ ( suc @ X3 ) @ A4 )
           => ( ( F2 @ ( suc @ X3 ) )
              = ( G2 @ ( suc @ X3 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F2 @ A4 )
          = ( groups6591440286371151544t_real @ G2 @ A4 ) ) ) ) ).

% sum_cong_Suc
thf(fact_5386_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: real > complex,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups5754745047067104278omplex @ G2
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X4: real] : ( if_complex @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5387_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G2: nat > complex,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2073611262835488442omplex @ G2
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X4: nat] : ( if_complex @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5388_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: int > complex,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3049146728041665814omplex @ G2
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X4: int] : ( if_complex @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5389_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G2: complex > complex,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex @ G2
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups7754918857620584856omplex
          @ ^ [X4: complex] : ( if_complex @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5390_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: real > real,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G2
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( if_real @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5391_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: int > real,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G2
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( if_real @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5392_sum_Ointer__filter,axiom,
    ! [A4: set_complex,G2: complex > real,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G2
          @ ( collect_complex
            @ ^ [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( if_real @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5393_sum_Ointer__filter,axiom,
    ! [A4: set_real,G2: real > rat,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat @ G2
          @ ( collect_real
            @ ^ [X4: real] :
                ( ( member_real @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X4: real] : ( if_rat @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5394_sum_Ointer__filter,axiom,
    ! [A4: set_nat,G2: nat > rat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat @ G2
          @ ( collect_nat
            @ ^ [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5395_sum_Ointer__filter,axiom,
    ! [A4: set_int,G2: int > rat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G2
          @ ( collect_int
            @ ^ [X4: int] :
                ( ( member_int @ X4 @ A4 )
                & ( P2 @ X4 ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P2 @ X4 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_filter
thf(fact_5396_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5397_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5398_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5399_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_5400_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ A ) )
        @ ( collect_complex
          @ ^ [C3: complex] : ( dvd_dvd_complex @ C3 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5401_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5402_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5403_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_5404_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G2: int > real,I2: int > int,F2: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ S2 ) @ ( groups8778361861064173332t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5405_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G2: complex > real,I2: complex > int,F2: int > real] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ S2 ) @ ( groups5808333547571424918x_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5406_sum__le__included,axiom,
    ! [S2: set_complex,T: set_int,G2: int > real,I2: int > complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ S2 ) @ ( groups8778361861064173332t_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5407_sum__le__included,axiom,
    ! [S2: set_complex,T: set_complex,G2: complex > real,I2: complex > complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X3 ) ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ S2 ) @ ( groups5808333547571424918x_real @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5408_sum__le__included,axiom,
    ! [S2: set_nat,T: set_nat,G2: nat > rat,I2: nat > nat,F2: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ S2 ) @ ( groups2906978787729119204at_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5409_sum__le__included,axiom,
    ! [S2: set_nat,T: set_int,G2: int > rat,I2: int > nat,F2: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5410_sum__le__included,axiom,
    ! [S2: set_nat,T: set_complex,G2: complex > rat,I2: complex > nat,F2: nat > rat] :
      ( ( finite_finite_nat @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5411_sum__le__included,axiom,
    ! [S2: set_int,T: set_nat,G2: nat > rat,I2: nat > int,F2: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_nat @ T )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ S2 ) @ ( groups2906978787729119204at_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5412_sum__le__included,axiom,
    ! [S2: set_int,T: set_int,G2: int > rat,I2: int > int,F2: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite_finite_int @ T )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ S2 ) @ ( groups3906332499630173760nt_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5413_sum__le__included,axiom,
    ! [S2: set_int,T: set_complex,G2: complex > rat,I2: complex > int,F2: int > rat] :
      ( ( finite_finite_int @ S2 )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G2 @ X3 ) ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S2 )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I2 @ Xa )
                      = X3 )
                    & ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ S2 ) @ ( groups5058264527183730370ex_rat @ G2 @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_5414_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F2 @ A4 )
            = zero_zero_real )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5415_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F2 @ A4 )
            = zero_zero_real )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5416_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F2 @ A4 )
            = zero_zero_real )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5417_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F2 @ A4 )
            = zero_zero_rat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5418_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F2 @ A4 )
            = zero_zero_rat )
          = ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5419_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F2 @ A4 )
            = zero_zero_rat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5420_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F2 @ A4 )
            = zero_zero_rat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5421_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_real,F2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F2 @ A4 )
            = zero_zero_nat )
          = ( ! [X4: real] :
                ( ( member_real @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5422_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F2 @ A4 )
            = zero_zero_nat )
          = ( ! [X4: int] :
                ( ( member_int @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5423_sum__nonneg__eq__0__iff,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F2 @ A4 )
            = zero_zero_nat )
          = ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A4 )
               => ( ( F2 @ X4 )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_5424_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: int > real,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_real @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ ( groups8778361861064173332t_real @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5425_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_real @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_real @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ ( groups5808333547571424918x_real @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5426_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: nat > rat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ( ord_less_rat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) @ ( groups2906978787729119204at_rat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5427_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_rat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ ( groups3906332499630173760nt_rat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5428_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_rat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_rat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ ( groups5058264527183730370ex_rat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5429_sum__strict__mono__ex1,axiom,
    ! [A4: set_int,F2: int > nat,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: int] :
              ( ( member_int @ X5 @ A4 )
              & ( ord_less_nat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( groups4541462559716669496nt_nat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5430_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: complex > nat,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_nat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( groups5693394587270226106ex_nat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5431_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: nat > int,G2: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ( ord_less_int @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_int @ ( groups3539618377306564664at_int @ F2 @ A4 ) @ ( groups3539618377306564664at_int @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5432_sum__strict__mono__ex1,axiom,
    ! [A4: set_complex,F2: complex > int,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ A4 )
           => ( ord_less_eq_int @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: complex] :
              ( ( member_complex @ X5 @ A4 )
              & ( ord_less_int @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_int @ ( groups5690904116761175830ex_int @ F2 @ A4 ) @ ( groups5690904116761175830ex_int @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5433_sum__strict__mono__ex1,axiom,
    ! [A4: set_nat,F2: nat > nat,G2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A4 )
           => ( ord_less_eq_nat @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) )
       => ( ? [X5: nat] :
              ( ( member_nat @ X5 @ A4 )
              & ( ord_less_nat @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) )
         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F2 @ A4 ) @ ( groups3542108847815614940at_nat @ G2 @ A4 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_5434_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_nat,H: nat > complex,G2: nat > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups2073611262835488442omplex @ H @ S3 ) @ ( groups2073611262835488442omplex @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5435_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_int,H: int > complex,G2: int > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups3049146728041665814omplex @ H @ S3 ) @ ( groups3049146728041665814omplex @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5436_sum_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( R2 @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_complex @ X15 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups7754918857620584856omplex @ H @ S3 ) @ ( groups7754918857620584856omplex @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5437_sum_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_int,H: int > real,G2: int > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups8778361861064173332t_real @ H @ S3 ) @ ( groups8778361861064173332t_real @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5438_sum_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_complex,H: complex > real,G2: complex > real] :
      ( ( R2 @ zero_zero_real @ zero_zero_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_real @ X15 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups5808333547571424918x_real @ H @ S3 ) @ ( groups5808333547571424918x_real @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5439_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_nat,H: nat > rat,G2: nat > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups2906978787729119204at_rat @ H @ S3 ) @ ( groups2906978787729119204at_rat @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5440_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_int,H: int > rat,G2: int > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups3906332499630173760nt_rat @ H @ S3 ) @ ( groups3906332499630173760nt_rat @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5441_sum_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_complex,H: complex > rat,G2: complex > rat] :
      ( ( R2 @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_rat @ X15 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups5058264527183730370ex_rat @ H @ S3 ) @ ( groups5058264527183730370ex_rat @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5442_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S3: set_int,H: int > nat,G2: int > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups4541462559716669496nt_nat @ H @ S3 ) @ ( groups4541462559716669496nt_nat @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5443_sum_Orelated,axiom,
    ! [R2: nat > nat > $o,S3: set_complex,H: complex > nat,G2: complex > nat] :
      ( ( R2 @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X15: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( plus_plus_nat @ X15 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups5693394587270226106ex_nat @ H @ S3 ) @ ( groups5693394587270226106ex_nat @ G2 @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_5444_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S3: set_real,I2: real > real,J2: real > real,T5: set_real,G2: real > complex,H: real > complex] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_real @ ( J2 @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups5754745047067104278omplex @ G2 @ S3 )
                        = ( groups5754745047067104278omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5445_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_int,S3: set_real,I2: int > real,J2: real > int,T5: set_int,G2: real > complex,H: int > complex] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_int @ ( J2 @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups5754745047067104278omplex @ G2 @ S3 )
                        = ( groups3049146728041665814omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5446_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_complex,S3: set_real,I2: complex > real,J2: real > complex,T5: set_complex,G2: real > complex,H: complex > complex] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_complex @ ( J2 @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups5754745047067104278omplex @ G2 @ S3 )
                        = ( groups7754918857620584856omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5447_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_real,S3: set_int,I2: real > int,J2: int > real,T5: set_real,G2: int > complex,H: real > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_real @ ( J2 @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups3049146728041665814omplex @ G2 @ S3 )
                        = ( groups5754745047067104278omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5448_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_int,S3: set_int,I2: int > int,J2: int > int,T5: set_int,G2: int > complex,H: int > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_int @ ( J2 @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups3049146728041665814omplex @ G2 @ S3 )
                        = ( groups3049146728041665814omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5449_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_int,T4: set_complex,S3: set_int,I2: complex > int,J2: int > complex,T5: set_complex,G2: int > complex,H: complex > complex] :
      ( ( finite_finite_int @ S5 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: int] :
              ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ S3 @ S5 ) )
               => ( member_complex @ ( J2 @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_int @ ( I2 @ B3 ) @ ( minus_minus_set_int @ S3 @ S5 ) ) )
               => ( ! [A3: int] :
                      ( ( member_int @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: int] :
                          ( ( member_int @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups3049146728041665814omplex @ G2 @ S3 )
                        = ( groups7754918857620584856omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5450_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_real,S3: set_complex,I2: real > complex,J2: complex > real,T5: set_real,G2: complex > complex,H: real > complex] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
               => ( member_real @ ( J2 @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S5 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups7754918857620584856omplex @ G2 @ S3 )
                        = ( groups5754745047067104278omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5451_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_int,S3: set_complex,I2: int > complex,J2: complex > int,T5: set_int,G2: complex > complex,H: int > complex] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite_finite_int @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
               => ( member_int @ ( J2 @ A3 ) @ ( minus_minus_set_int @ T5 @ T4 ) ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: int] :
                    ( ( member_int @ B3 @ ( minus_minus_set_int @ T5 @ T4 ) )
                   => ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S5 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: int] :
                        ( ( member_int @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups7754918857620584856omplex @ G2 @ S3 )
                        = ( groups3049146728041665814omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5452_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_complex,T4: set_complex,S3: set_complex,I2: complex > complex,J2: complex > complex,T5: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ S5 )
     => ( ( finite3207457112153483333omplex @ T4 )
       => ( ! [A3: complex] :
              ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ S3 @ S5 ) )
               => ( member_complex @ ( J2 @ A3 ) @ ( minus_811609699411566653omplex @ T5 @ T4 ) ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: complex] :
                    ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ T5 @ T4 ) )
                   => ( member_complex @ ( I2 @ B3 ) @ ( minus_811609699411566653omplex @ S3 @ S5 ) ) )
               => ( ! [A3: complex] :
                      ( ( member_complex @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_complex ) )
                 => ( ! [B3: complex] :
                        ( ( member_complex @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_complex ) )
                   => ( ! [A3: complex] :
                          ( ( member_complex @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups7754918857620584856omplex @ G2 @ S3 )
                        = ( groups7754918857620584856omplex @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5453_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S5: set_real,T4: set_real,S3: set_real,I2: real > real,J2: real > real,T5: set_real,G2: real > real,H: real > real] :
      ( ( finite_finite_real @ S5 )
     => ( ( finite_finite_real @ T4 )
       => ( ! [A3: real] :
              ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
             => ( ( I2 @ ( J2 @ A3 ) )
                = A3 ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ S3 @ S5 ) )
               => ( member_real @ ( J2 @ A3 ) @ ( minus_minus_set_real @ T5 @ T4 ) ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                 => ( ( J2 @ ( I2 @ B3 ) )
                    = B3 ) )
             => ( ! [B3: real] :
                    ( ( member_real @ B3 @ ( minus_minus_set_real @ T5 @ T4 ) )
                   => ( member_real @ ( I2 @ B3 ) @ ( minus_minus_set_real @ S3 @ S5 ) ) )
               => ( ! [A3: real] :
                      ( ( member_real @ A3 @ S5 )
                     => ( ( G2 @ A3 )
                        = zero_zero_real ) )
                 => ( ! [B3: real] :
                        ( ( member_real @ B3 @ T4 )
                       => ( ( H @ B3 )
                          = zero_zero_real ) )
                   => ( ! [A3: real] :
                          ( ( member_real @ A3 @ S3 )
                         => ( ( H @ ( J2 @ A3 ) )
                            = ( G2 @ A3 ) ) )
                     => ( ( groups8097168146408367636l_real @ G2 @ S3 )
                        = ( groups8097168146408367636l_real @ H @ T5 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_5454_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_5455_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_5456_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_5457_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5458_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5459_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5460_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5461_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_5462_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_5463_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_5464_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_mult_right_cancel
thf(fact_5465_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C2 ) )
        = ( B = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_5466_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C2 ) )
        = ( B = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_5467_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C2 ) )
        = ( B = C2 ) ) ) ).

% unit_mult_left_cancel
thf(fact_5468_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C2 )
        = ( dvd_dvd_Code_integer @ B @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5469_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
        = ( dvd_dvd_nat @ B @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5470_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
        = ( dvd_dvd_int @ B @ C2 ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_5471_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
        = ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5472_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5473_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_5474_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C2 )
        = ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5475_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C2 )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5476_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C2 )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% mult_unit_dvd_iff
thf(fact_5477_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C2 @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5478_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5479_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_mult_unit_iff
thf(fact_5480_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_5481_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_5482_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_5483_dvd__div__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C2 ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_5484_dvd__div__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C2 ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C2 ) ) ) ).

% dvd_div_mult
thf(fact_5485_div__mult__swap,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_5486_div__mult__swap,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 ) ) ) ).

% div_mult_swap
thf(fact_5487_div__div__eq__right,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_5488_div__div__eq__right,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C2 ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).

% div_div_eq_right
thf(fact_5489_dvd__div__mult2__eq,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C2 ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_5490_dvd__div__mult2__eq,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C2 ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ).

% dvd_div_mult2_eq
thf(fact_5491_dvd__mult__imp__div,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_5492_dvd__mult__imp__div,axiom,
    ! [A: int,C2: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C2 ) ) ) ).

% dvd_mult_imp_div
thf(fact_5493_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C2 )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C2 @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_5494_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C2: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C2 )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C2 @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_5495_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_div_cancel
thf(fact_5496_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_div_cancel
thf(fact_5497_unit__div__cancel,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C2 @ A ) )
        = ( B = C2 ) ) ) ).

% unit_div_cancel
thf(fact_5498_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C2 )
        = ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_5499_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_5500_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% div_unit_dvd_iff
thf(fact_5501_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C2 @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_5502_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C2 @ B ) )
        = ( dvd_dvd_nat @ A @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_5503_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C2 @ B ) )
        = ( dvd_dvd_int @ A @ C2 ) ) ) ).

% dvd_div_unit_iff
thf(fact_5504_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5505_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5506_mod__eq__0__iff__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural )
      = ( dvd_dvd_Code_natural @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_5507_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( modulo_modulo_nat @ B4 @ A5 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5508_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A5: int,B4: int] :
          ( ( modulo_modulo_int @ B4 @ A5 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5509_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_natural
    = ( ^ [A5: code_natural,B4: code_natural] :
          ( ( modulo8411746178871703098atural @ B4 @ A5 )
          = zero_z2226904508553997617atural ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_5510_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5511_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5512_mod__0__imp__dvd,axiom,
    ! [A: code_natural,B: code_natural] :
      ( ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural )
     => ( dvd_dvd_Code_natural @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_5513_dvd__power__le,axiom,
    ! [X: nat,Y: nat,N: nat,M2: nat] :
      ( ( dvd_dvd_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5514_dvd__power__le,axiom,
    ! [X: real,Y: real,N: nat,M2: nat] :
      ( ( dvd_dvd_real @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5515_dvd__power__le,axiom,
    ! [X: int,Y: int,N: nat,M2: nat] :
      ( ( dvd_dvd_int @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5516_dvd__power__le,axiom,
    ! [X: complex,Y: complex,N: nat,M2: nat] :
      ( ( dvd_dvd_complex @ X @ Y )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ M2 ) ) ) ) ).

% dvd_power_le
thf(fact_5517_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_5518_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_5519_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_5520_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_5521_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_5522_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_5523_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_5524_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_5525_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5526_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5527_dvd__minus__mod,axiom,
    ! [B: code_natural,A: code_natural] : ( dvd_dvd_Code_natural @ B @ ( minus_7197305767214868737atural @ A @ ( modulo8411746178871703098atural @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_5528_dvd__pos__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ M2 @ N )
       => ( ord_less_nat @ zero_zero_nat @ M2 ) ) ) ).

% dvd_pos_nat
thf(fact_5529_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_5530_pochhammer__pos,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_5531_pochhammer__pos,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_5532_pochhammer__pos,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_5533_pochhammer__pos,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_pos
thf(fact_5534_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_5535_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: real > real,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F2 @ S2 )
            = zero_zero_real )
         => ( ( member_real @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5536_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: int > real,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F2 @ S2 )
            = zero_zero_real )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5537_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: complex > real,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F2 @ S2 )
            = zero_zero_real )
         => ( ( member_complex @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5538_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: real > rat,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F2 @ S2 )
            = zero_zero_rat )
         => ( ( member_real @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5539_sum__nonneg__0,axiom,
    ! [S2: set_nat,F2: nat > rat,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F2 @ S2 )
            = zero_zero_rat )
         => ( ( member_nat @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5540_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: int > rat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F2 @ S2 )
            = zero_zero_rat )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5541_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: complex > rat,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F2 @ S2 )
            = zero_zero_rat )
         => ( ( member_complex @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5542_sum__nonneg__0,axiom,
    ! [S2: set_real,F2: real > nat,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F2 @ S2 )
            = zero_zero_nat )
         => ( ( member_real @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5543_sum__nonneg__0,axiom,
    ! [S2: set_int,F2: int > nat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F2 @ S2 )
            = zero_zero_nat )
         => ( ( member_int @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5544_sum__nonneg__0,axiom,
    ! [S2: set_complex,F2: complex > nat,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F2 @ S2 )
            = zero_zero_nat )
         => ( ( member_complex @ I2 @ S2 )
           => ( ( F2 @ I2 )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_5545_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: real > real,B5: real,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F2 @ S2 )
            = B5 )
         => ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_real @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5546_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: int > real,B5: real,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F2 @ S2 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_real @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5547_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: complex > real,B5: real,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F2 @ S2 )
            = B5 )
         => ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_real @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5548_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: real > rat,B5: rat,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F2 @ S2 )
            = B5 )
         => ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5549_sum__nonneg__leq__bound,axiom,
    ! [S2: set_nat,F2: nat > rat,B5: rat,I2: nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F2 @ S2 )
            = B5 )
         => ( ( member_nat @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5550_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: int > rat,B5: rat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F2 @ S2 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5551_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: complex > rat,B5: rat,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F2 @ S2 )
            = B5 )
         => ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_rat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5552_sum__nonneg__leq__bound,axiom,
    ! [S2: set_real,F2: real > nat,B5: nat,I2: real] :
      ( ( finite_finite_real @ S2 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F2 @ S2 )
            = B5 )
         => ( ( member_real @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5553_sum__nonneg__leq__bound,axiom,
    ! [S2: set_int,F2: int > nat,B5: nat,I2: int] :
      ( ( finite_finite_int @ S2 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F2 @ S2 )
            = B5 )
         => ( ( member_int @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5554_sum__nonneg__leq__bound,axiom,
    ! [S2: set_complex,F2: complex > nat,B5: nat,I2: complex] :
      ( ( finite3207457112153483333omplex @ S2 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F2 @ S2 )
            = B5 )
         => ( ( member_complex @ I2 @ S2 )
           => ( ord_less_eq_nat @ ( F2 @ I2 ) @ B5 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_5555_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_5556_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_5557_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_5558_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N: nat,M2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ M2 )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_5559_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_5560_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_5561_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M2: nat,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( ( comm_s2602460028002588243omplex @ A @ N )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_5562_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_5563_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_5564_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) )
        | ( ( times_times_nat @ B @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_nat
thf(fact_5565_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X3: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X3 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_5566_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D5: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D5 @ A )
      & ( dvd_dvd_nat @ D5 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = D5 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = D5 ) ) ) ).

% bezout1_nat
thf(fact_5567_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: real > complex,B5: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups5754745047067104278omplex @ G2 @ ( inf_inf_set_real @ A4 @ B5 ) )
        = ( groups5754745047067104278omplex
          @ ^ [X4: real] : ( if_complex @ ( member_real @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5568_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: int > complex,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3049146728041665814omplex @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups3049146728041665814omplex
          @ ^ [X4: int] : ( if_complex @ ( member_int @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5569_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: complex > complex,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) )
        = ( groups7754918857620584856omplex
          @ ^ [X4: complex] : ( if_complex @ ( member_complex @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_complex )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5570_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: real > real,B5: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G2 @ ( inf_inf_set_real @ A4 @ B5 ) )
        = ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( if_real @ ( member_real @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5571_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: int > real,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( if_real @ ( member_int @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5572_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: complex > real,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) )
        = ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( if_real @ ( member_complex @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_real )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5573_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: real > rat,B5: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat @ G2 @ ( inf_inf_set_real @ A4 @ B5 ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X4: real] : ( if_rat @ ( member_real @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5574_sum_Ointer__restrict,axiom,
    ! [A4: set_int,G2: int > rat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( if_rat @ ( member_int @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5575_sum_Ointer__restrict,axiom,
    ! [A4: set_complex,G2: complex > rat,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X4: complex] : ( if_rat @ ( member_complex @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_rat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5576_sum_Ointer__restrict,axiom,
    ! [A4: set_real,G2: real > nat,B5: set_real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1935376822645274424al_nat @ G2 @ ( inf_inf_set_real @ A4 @ B5 ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( if_nat @ ( member_real @ X4 @ B5 ) @ ( G2 @ X4 ) @ zero_zero_nat )
          @ A4 ) ) ) ).

% sum.inter_restrict
thf(fact_5577_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: real > complex] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups5754745047067104278omplex @ G2
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G2 @ X4 )
                  = zero_zero_complex ) ) ) )
        = ( groups5754745047067104278omplex @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5578_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3049146728041665814omplex @ G2
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G2 @ X4 )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5579_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups7754918857620584856omplex @ G2
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G2 @ X4 )
                  = zero_zero_complex ) ) ) )
        = ( groups7754918857620584856omplex @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5580_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real @ G2
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G2 @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5581_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real @ G2
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G2 @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5582_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real @ G2
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G2 @ X4 )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5583_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat @ G2
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G2 @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5584_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat @ G2
          @ ( minus_minus_set_int @ A4
            @ ( collect_int
              @ ^ [X4: int] :
                  ( ( G2 @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5585_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat @ G2
          @ ( minus_811609699411566653omplex @ A4
            @ ( collect_complex
              @ ^ [X4: complex] :
                  ( ( G2 @ X4 )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5586_sum_Osetdiff__irrelevant,axiom,
    ! [A4: set_real,G2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1935376822645274424al_nat @ G2
          @ ( minus_minus_set_real @ A4
            @ ( collect_real
              @ ^ [X4: real] :
                  ( ( G2 @ X4 )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G2 @ A4 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_5587_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5588_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_5589_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G2 @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5590_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > real,M2: nat,K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G2 @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_5591_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5592_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F2 @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5593_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I2 @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F2 @ I2 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5594_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5595_sum__pos2,axiom,
    ! [I5: set_nat,I2: nat,F2: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5596_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5597_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I2 @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ I2 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5598_sum__pos2,axiom,
    ! [I5: set_real,I2: real,F2: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5599_sum__pos2,axiom,
    ! [I5: set_int,I2: int,F2: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5600_sum__pos2,axiom,
    ! [I5: set_complex,I2: complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I2 @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ I2 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F2 @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_5601_sum__pos,axiom,
    ! [I5: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F2 @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5602_sum__pos,axiom,
    ! [I5: set_int,F2: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F2 @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5603_sum__pos,axiom,
    ! [I5: set_real,F2: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F2 @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5604_sum__pos,axiom,
    ! [I5: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5605_sum__pos,axiom,
    ! [I5: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5606_sum__pos,axiom,
    ! [I5: set_int,F2: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5607_sum__pos,axiom,
    ! [I5: set_real,F2: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F2 @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5608_sum__pos,axiom,
    ! [I5: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5609_sum__pos,axiom,
    ! [I5: set_int,F2: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5610_sum__pos,axiom,
    ! [I5: set_real,F2: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F2 @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_5611_sum__bounded__below,axiom,
    ! [A4: set_real,K5: real,F2: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_real @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) @ K5 ) @ ( groups8097168146408367636l_real @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5612_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: real,F2: complex > real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_real @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ K5 ) @ ( groups5808333547571424918x_real @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5613_sum__bounded__below,axiom,
    ! [A4: set_int,K5: real,F2: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_real @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ K5 ) @ ( groups8778361861064173332t_real @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5614_sum__bounded__below,axiom,
    ! [A4: set_real,K5: rat,F2: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_rat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A4 ) ) @ K5 ) @ ( groups1300246762558778688al_rat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5615_sum__bounded__below,axiom,
    ! [A4: set_nat,K5: rat,F2: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_eq_rat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) @ K5 ) @ ( groups2906978787729119204at_rat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5616_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: rat,F2: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_rat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A4 ) ) @ K5 ) @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5617_sum__bounded__below,axiom,
    ! [A4: set_int,K5: rat,F2: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_rat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) @ K5 ) @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5618_sum__bounded__below,axiom,
    ! [A4: set_real,K5: nat,F2: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_nat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A4 ) ) @ K5 ) @ ( groups1935376822645274424al_nat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5619_sum__bounded__below,axiom,
    ! [A4: set_complex,K5: nat,F2: complex > nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_nat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A4 ) ) @ K5 ) @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5620_sum__bounded__below,axiom,
    ! [A4: set_int,K5: nat,F2: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_nat @ K5 @ ( F2 @ I3 ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A4 ) ) @ K5 ) @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) ) ) ).

% sum_bounded_below
thf(fact_5621_sum__bounded__above,axiom,
    ! [A4: set_real,F2: real > real,K5: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5622_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: complex > real,K5: real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5623_sum__bounded__above,axiom,
    ! [A4: set_int,F2: int > real,K5: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5624_sum__bounded__above,axiom,
    ! [A4: set_real,F2: real > rat,K5: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5625_sum__bounded__above,axiom,
    ! [A4: set_nat,F2: nat > rat,K5: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5626_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: complex > rat,K5: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5627_sum__bounded__above,axiom,
    ! [A4: set_int,F2: int > rat,K5: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5628_sum__bounded__above,axiom,
    ! [A4: set_real,F2: real > nat,K5: nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5629_sum__bounded__above,axiom,
    ! [A4: set_complex,F2: complex > nat,K5: nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5630_sum__bounded__above,axiom,
    ! [A4: set_int,F2: int > nat,K5: nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I3 ) @ K5 ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( finite_card_int @ A4 ) ) @ K5 ) ) ) ).

% sum_bounded_above
thf(fact_5631_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G2: real > complex,H: real > complex] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G2 @ T5 )
              = ( groups5754745047067104278omplex @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5632_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > complex,H: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G2 @ T5 )
              = ( groups3049146728041665814omplex @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5633_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7754918857620584856omplex @ G2 @ T5 )
              = ( groups7754918857620584856omplex @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5634_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G2: real > real,H: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G2 @ T5 )
              = ( groups8097168146408367636l_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5635_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > real,H: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G2 @ T5 )
              = ( groups8778361861064173332t_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5636_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G2 @ T5 )
              = ( groups5808333547571424918x_real @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5637_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G2: real > rat,H: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G2 @ T5 )
              = ( groups1300246762558778688al_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5638_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > rat,H: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G2 @ T5 )
              = ( groups3906332499630173760nt_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5639_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G2 @ T5 )
              = ( groups5058264527183730370ex_rat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5640_sum_Omono__neutral__cong__right,axiom,
    ! [T5: set_real,S3: set_real,G2: real > nat,H: real > nat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1935376822645274424al_nat @ G2 @ T5 )
              = ( groups1935376822645274424al_nat @ H @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_5641_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: real > complex,G2: real > complex] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5754745047067104278omplex @ G2 @ S3 )
              = ( groups5754745047067104278omplex @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5642_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H: int > complex,G2: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3049146728041665814omplex @ G2 @ S3 )
              = ( groups3049146728041665814omplex @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5643_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_complex ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups7754918857620584856omplex @ G2 @ S3 )
              = ( groups7754918857620584856omplex @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5644_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: real > real,G2: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8097168146408367636l_real @ G2 @ S3 )
              = ( groups8097168146408367636l_real @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5645_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H: int > real,G2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups8778361861064173332t_real @ G2 @ S3 )
              = ( groups8778361861064173332t_real @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5646_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_real ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5808333547571424918x_real @ G2 @ S3 )
              = ( groups5808333547571424918x_real @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5647_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1300246762558778688al_rat @ G2 @ S3 )
              = ( groups1300246762558778688al_rat @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5648_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_int,S3: set_int,H: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups3906332499630173760nt_rat @ G2 @ S3 )
              = ( groups3906332499630173760nt_rat @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5649_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_rat ) )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G2 @ S3 )
              = ( groups5058264527183730370ex_rat @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5650_sum_Omono__neutral__cong__left,axiom,
    ! [T5: set_real,S3: set_real,H: real > nat,G2: real > nat] :
      ( ( finite_finite_real @ T5 )
     => ( ( ord_less_eq_set_real @ S3 @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ X3 )
                = zero_zero_nat ) )
         => ( ! [X3: real] :
                ( ( member_real @ X3 @ S3 )
               => ( ( G2 @ X3 )
                  = ( H @ X3 ) ) )
           => ( ( groups1935376822645274424al_nat @ G2 @ S3 )
              = ( groups1935376822645274424al_nat @ H @ T5 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_5651_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G2 @ T5 )
            = ( groups3049146728041665814omplex @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5652_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G2 @ T5 )
            = ( groups7754918857620584856omplex @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5653_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G2 @ T5 )
            = ( groups8778361861064173332t_real @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5654_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G2 @ T5 )
            = ( groups5808333547571424918x_real @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5655_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G2 @ T5 )
            = ( groups3906332499630173760nt_rat @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5656_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G2 @ T5 )
            = ( groups5058264527183730370ex_rat @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5657_sum_Omono__neutral__right,axiom,
    ! [T5: set_int,S3: set_int,G2: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G2 @ T5 )
            = ( groups4541462559716669496nt_nat @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5658_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G2 @ T5 )
            = ( groups5693394587270226106ex_nat @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5659_sum_Omono__neutral__right,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G2 @ T5 )
            = ( groups5690904116761175830ex_int @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5660_sum_Omono__neutral__right,axiom,
    ! [T5: set_nat,S3: set_nat,G2: nat > complex] :
      ( ( finite_finite_nat @ T5 )
     => ( ( ord_less_eq_set_nat @ S3 @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G2 @ T5 )
            = ( groups2073611262835488442omplex @ G2 @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_5661_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G2: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G2 @ S3 )
            = ( groups3049146728041665814omplex @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5662_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G2 @ S3 )
            = ( groups7754918857620584856omplex @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5663_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G2 @ S3 )
            = ( groups8778361861064173332t_real @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5664_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G2 @ S3 )
            = ( groups5808333547571424918x_real @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5665_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G2 @ S3 )
            = ( groups3906332499630173760nt_rat @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5666_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G2 @ S3 )
            = ( groups5058264527183730370ex_rat @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5667_sum_Omono__neutral__left,axiom,
    ! [T5: set_int,S3: set_int,G2: int > nat] :
      ( ( finite_finite_int @ T5 )
     => ( ( ord_less_eq_set_int @ S3 @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G2 @ S3 )
            = ( groups4541462559716669496nt_nat @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5668_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G2 @ S3 )
            = ( groups5693394587270226106ex_nat @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5669_sum_Omono__neutral__left,axiom,
    ! [T5: set_complex,S3: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G2 @ S3 )
            = ( groups5690904116761175830ex_int @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5670_sum_Omono__neutral__left,axiom,
    ! [T5: set_nat,S3: set_nat,G2: nat > complex] :
      ( ( finite_finite_nat @ T5 )
     => ( ( ord_less_eq_set_nat @ S3 @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( minus_minus_set_nat @ T5 @ S3 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G2 @ S3 )
            = ( groups2073611262835488442omplex @ G2 @ T5 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_5671_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > complex,H: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G2 @ C4 )
                  = ( groups5754745047067104278omplex @ H @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G2 @ A4 )
                  = ( groups5754745047067104278omplex @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5672_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > complex,H: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G2 @ C4 )
                  = ( groups3049146728041665814omplex @ H @ C4 ) )
               => ( ( groups3049146728041665814omplex @ G2 @ A4 )
                  = ( groups3049146728041665814omplex @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5673_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups7754918857620584856omplex @ G2 @ C4 )
                  = ( groups7754918857620584856omplex @ H @ C4 ) )
               => ( ( groups7754918857620584856omplex @ G2 @ A4 )
                  = ( groups7754918857620584856omplex @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5674_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > real,H: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G2 @ C4 )
                  = ( groups8097168146408367636l_real @ H @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G2 @ A4 )
                  = ( groups8097168146408367636l_real @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5675_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > real,H: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G2 @ C4 )
                  = ( groups8778361861064173332t_real @ H @ C4 ) )
               => ( ( groups8778361861064173332t_real @ G2 @ A4 )
                  = ( groups8778361861064173332t_real @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5676_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G2 @ C4 )
                  = ( groups5808333547571424918x_real @ H @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G2 @ A4 )
                  = ( groups5808333547571424918x_real @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5677_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G2 @ C4 )
                  = ( groups1300246762558778688al_rat @ H @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G2 @ A4 )
                  = ( groups1300246762558778688al_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5678_sum_Osame__carrierI,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > rat,H: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G2 @ C4 )
                  = ( groups3906332499630173760nt_rat @ H @ C4 ) )
               => ( ( groups3906332499630173760nt_rat @ G2 @ A4 )
                  = ( groups3906332499630173760nt_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5679_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G2 @ C4 )
                  = ( groups5058264527183730370ex_rat @ H @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G2 @ A4 )
                  = ( groups5058264527183730370ex_rat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5680_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > nat,H: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G2 @ C4 )
                  = ( groups1935376822645274424al_nat @ H @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G2 @ A4 )
                  = ( groups1935376822645274424al_nat @ H @ B5 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_5681_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > complex,H: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G2 @ A4 )
                  = ( groups5754745047067104278omplex @ H @ B5 ) )
                = ( ( groups5754745047067104278omplex @ G2 @ C4 )
                  = ( groups5754745047067104278omplex @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5682_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > complex,H: int > complex] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups3049146728041665814omplex @ G2 @ A4 )
                  = ( groups3049146728041665814omplex @ H @ B5 ) )
                = ( ( groups3049146728041665814omplex @ G2 @ C4 )
                  = ( groups3049146728041665814omplex @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5683_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > complex,H: complex > complex] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_complex ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_complex ) )
             => ( ( ( groups7754918857620584856omplex @ G2 @ A4 )
                  = ( groups7754918857620584856omplex @ H @ B5 ) )
                = ( ( groups7754918857620584856omplex @ G2 @ C4 )
                  = ( groups7754918857620584856omplex @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5684_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > real,H: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G2 @ A4 )
                  = ( groups8097168146408367636l_real @ H @ B5 ) )
                = ( ( groups8097168146408367636l_real @ G2 @ C4 )
                  = ( groups8097168146408367636l_real @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5685_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > real,H: int > real] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups8778361861064173332t_real @ G2 @ A4 )
                  = ( groups8778361861064173332t_real @ H @ B5 ) )
                = ( ( groups8778361861064173332t_real @ G2 @ C4 )
                  = ( groups8778361861064173332t_real @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5686_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > real,H: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_real ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G2 @ A4 )
                  = ( groups5808333547571424918x_real @ H @ B5 ) )
                = ( ( groups5808333547571424918x_real @ G2 @ C4 )
                  = ( groups5808333547571424918x_real @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5687_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > rat,H: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G2 @ A4 )
                  = ( groups1300246762558778688al_rat @ H @ B5 ) )
                = ( ( groups1300246762558778688al_rat @ G2 @ C4 )
                  = ( groups1300246762558778688al_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5688_sum_Osame__carrier,axiom,
    ! [C4: set_int,A4: set_int,B5: set_int,G2: int > rat,H: int > rat] :
      ( ( finite_finite_int @ C4 )
     => ( ( ord_less_eq_set_int @ A4 @ C4 )
       => ( ( ord_less_eq_set_int @ B5 @ C4 )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ ( minus_minus_set_int @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: int] :
                  ( ( member_int @ B3 @ ( minus_minus_set_int @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups3906332499630173760nt_rat @ G2 @ A4 )
                  = ( groups3906332499630173760nt_rat @ H @ B5 ) )
                = ( ( groups3906332499630173760nt_rat @ G2 @ C4 )
                  = ( groups3906332499630173760nt_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5689_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A4: set_complex,B5: set_complex,G2: complex > rat,H: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A4 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B5 @ C4 )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ ( minus_811609699411566653omplex @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_rat ) )
           => ( ! [B3: complex] :
                  ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G2 @ A4 )
                  = ( groups5058264527183730370ex_rat @ H @ B5 ) )
                = ( ( groups5058264527183730370ex_rat @ G2 @ C4 )
                  = ( groups5058264527183730370ex_rat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5690_sum_Osame__carrier,axiom,
    ! [C4: set_real,A4: set_real,B5: set_real,G2: real > nat,H: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A4 @ C4 )
       => ( ( ord_less_eq_set_real @ B5 @ C4 )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ ( minus_minus_set_real @ C4 @ A4 ) )
               => ( ( G2 @ A3 )
                  = zero_zero_nat ) )
           => ( ! [B3: real] :
                  ( ( member_real @ B3 @ ( minus_minus_set_real @ C4 @ B5 ) )
                 => ( ( H @ B3 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G2 @ A4 )
                  = ( groups1935376822645274424al_nat @ H @ B5 ) )
                = ( ( groups1935376822645274424al_nat @ G2 @ C4 )
                  = ( groups1935376822645274424al_nat @ H @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_5691_sum_Omono__neutral__cong,axiom,
    ! [T5: set_real,S3: set_real,H: real > complex,G2: real > complex] :
      ( ( finite_finite_real @ T5 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_complex ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ ( minus_minus_set_real @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_complex ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ ( inf_inf_set_real @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups5754745047067104278omplex @ G2 @ S3 )
                = ( groups5754745047067104278omplex @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5692_sum_Omono__neutral__cong,axiom,
    ! [T5: set_int,S3: set_int,H: int > complex,G2: int > complex] :
      ( ( finite_finite_int @ T5 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_complex ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_complex ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3049146728041665814omplex @ G2 @ S3 )
                = ( groups3049146728041665814omplex @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5693_sum_Omono__neutral__cong,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( finite3207457112153483333omplex @ S3 )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_complex ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_complex ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ ( inf_inf_set_complex @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups7754918857620584856omplex @ G2 @ S3 )
                = ( groups7754918857620584856omplex @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5694_sum_Omono__neutral__cong,axiom,
    ! [T5: set_real,S3: set_real,H: real > real,G2: real > real] :
      ( ( finite_finite_real @ T5 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_real ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ ( minus_minus_set_real @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_real ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ ( inf_inf_set_real @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups8097168146408367636l_real @ G2 @ S3 )
                = ( groups8097168146408367636l_real @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5695_sum_Omono__neutral__cong,axiom,
    ! [T5: set_int,S3: set_int,H: int > real,G2: int > real] :
      ( ( finite_finite_int @ T5 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_real ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_real ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups8778361861064173332t_real @ G2 @ S3 )
                = ( groups8778361861064173332t_real @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5696_sum_Omono__neutral__cong,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( finite3207457112153483333omplex @ S3 )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_real ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_real ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ ( inf_inf_set_complex @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups5808333547571424918x_real @ G2 @ S3 )
                = ( groups5808333547571424918x_real @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5697_sum_Omono__neutral__cong,axiom,
    ! [T5: set_real,S3: set_real,H: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ T5 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ ( minus_minus_set_real @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ ( inf_inf_set_real @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups1300246762558778688al_rat @ G2 @ S3 )
                = ( groups1300246762558778688al_rat @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5698_sum_Omono__neutral__cong,axiom,
    ! [T5: set_int,S3: set_int,H: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ T5 )
     => ( ( finite_finite_int @ S3 )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ ( minus_minus_set_int @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ ( minus_minus_set_int @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ ( inf_inf_set_int @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups3906332499630173760nt_rat @ G2 @ S3 )
                = ( groups3906332499630173760nt_rat @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5699_sum_Omono__neutral__cong,axiom,
    ! [T5: set_complex,S3: set_complex,H: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T5 )
     => ( ( finite3207457112153483333omplex @ S3 )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_rat ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ ( minus_811609699411566653omplex @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_rat ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ ( inf_inf_set_complex @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups5058264527183730370ex_rat @ G2 @ S3 )
                = ( groups5058264527183730370ex_rat @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5700_sum_Omono__neutral__cong,axiom,
    ! [T5: set_real,S3: set_real,H: real > nat,G2: real > nat] :
      ( ( finite_finite_real @ T5 )
     => ( ( finite_finite_real @ S3 )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ ( minus_minus_set_real @ T5 @ S3 ) )
             => ( ( H @ I3 )
                = zero_zero_nat ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ ( minus_minus_set_real @ S3 @ T5 ) )
               => ( ( G2 @ I3 )
                  = zero_zero_nat ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ ( inf_inf_set_real @ S3 @ T5 ) )
                 => ( ( G2 @ X3 )
                    = ( H @ X3 ) ) )
             => ( ( groups1935376822645274424al_nat @ G2 @ S3 )
                = ( groups1935376822645274424al_nat @ H @ T5 ) ) ) ) ) ) ) ).

% sum.mono_neutral_cong
thf(fact_5701_unity__coeff__ex,axiom,
    ! [P2: complex > $o,L: complex] :
      ( ( ? [X4: complex] : ( P2 @ ( times_times_complex @ L @ X4 ) ) )
      = ( ? [X4: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X4 @ zero_zero_complex ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5702_unity__coeff__ex,axiom,
    ! [P2: real > $o,L: real] :
      ( ( ? [X4: real] : ( P2 @ ( times_times_real @ L @ X4 ) ) )
      = ( ? [X4: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X4 @ zero_zero_real ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5703_unity__coeff__ex,axiom,
    ! [P2: rat > $o,L: rat] :
      ( ( ? [X4: rat] : ( P2 @ ( times_times_rat @ L @ X4 ) ) )
      = ( ? [X4: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X4 @ zero_zero_rat ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5704_unity__coeff__ex,axiom,
    ! [P2: nat > $o,L: nat] :
      ( ( ? [X4: nat] : ( P2 @ ( times_times_nat @ L @ X4 ) ) )
      = ( ? [X4: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X4 @ zero_zero_nat ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5705_unity__coeff__ex,axiom,
    ! [P2: int > $o,L: int] :
      ( ( ? [X4: int] : ( P2 @ ( times_times_int @ L @ X4 ) ) )
      = ( ? [X4: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X4 @ zero_zero_int ) )
            & ( P2 @ X4 ) ) ) ) ).

% unity_coeff_ex
thf(fact_5706_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C ) ) ) ) ).

% unit_dvdE
thf(fact_5707_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C: nat] :
              ( B
             != ( times_times_nat @ A @ C ) ) ) ) ).

% unit_dvdE
thf(fact_5708_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C: int] :
              ( B
             != ( times_times_int @ A @ C ) ) ) ) ).

% unit_dvdE
thf(fact_5709_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C2: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C2 != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C2 @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C2 ) )
              = ( ( times_times_nat @ B @ C2 )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5710_dvd__div__div__eq__mult,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C2 != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C2 @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C2 ) )
              = ( ( times_times_int @ B @ C2 )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_5711_dvd__div__iff__mult,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( C2 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C2 @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5712_dvd__div__iff__mult,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( C2 != zero_zero_int )
     => ( ( dvd_dvd_int @ C2 @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C2 ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C2 ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_5713_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C2 @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5714_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C2 )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C2 @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_5715_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C2 )
          = ( B
            = ( times_times_nat @ C2 @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5716_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C2: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C2 )
          = ( B
            = ( times_times_int @ C2 @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_5717_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5718_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5719_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_5720_inf__period_I4_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( ( dvd_dvd_real @ D @ D3 )
     => ! [X5: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5721_inf__period_I4_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D3 )
     => ! [X5: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5722_inf__period_I4_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X5: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_5723_inf__period_I3_J,axiom,
    ! [D: real,D3: real,T: real] :
      ( ( dvd_dvd_real @ D @ D3 )
     => ! [X5: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X5 @ T ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X5 @ ( times_times_real @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5724_inf__period_I3_J,axiom,
    ! [D: rat,D3: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D3 )
     => ! [X5: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X5 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X5 @ ( times_times_rat @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5725_inf__period_I3_J,axiom,
    ! [D: int,D3: int,T: int] :
      ( ( dvd_dvd_int @ D @ D3 )
     => ! [X5: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X5 @ T ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X5 @ ( times_times_int @ K4 @ D3 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_5726_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C2 )
        = ( A
          = ( times_3573771949741848930nteger @ C2 @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_5727_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C2 )
        = ( A
          = ( times_times_nat @ C2 @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_5728_unit__eq__div1,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C2 )
        = ( A
          = ( times_times_int @ C2 @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_5729_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C2 @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_5730_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C2 @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_5731_unit__eq__div2,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C2 @ B ) )
        = ( ( times_times_int @ A @ B )
          = C2 ) ) ) ).

% unit_eq_div2
thf(fact_5732_div__mult__unit2,axiom,
    ! [C2: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_5733_div__mult__unit2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_5734_div__mult__unit2,axiom,
    ! [C2: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).

% div_mult_unit2
thf(fact_5735_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C2 ) @ B ) ) ) ).

% unit_div_commute
thf(fact_5736_unit__div__commute,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C2 )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C2 ) @ B ) ) ) ).

% unit_div_commute
thf(fact_5737_unit__div__commute,axiom,
    ! [B: int,A: int,C2: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C2 )
        = ( divide_divide_int @ ( times_times_int @ A @ C2 ) @ B ) ) ) ).

% unit_div_commute
thf(fact_5738_unit__div__mult__swap,axiom,
    ! [C2: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C2 ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_5739_unit__div__mult__swap,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C2 @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C2 ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_5740_unit__div__mult__swap,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C2 @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C2 ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C2 ) ) ) ).

% unit_div_mult_swap
thf(fact_5741_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C2: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C2 @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C2 ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5742_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C2 @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C2 ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5743_is__unit__div__mult2__eq,axiom,
    ! [B: int,C2: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C2 @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C2 ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C2 ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_5744_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_5745_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_5746_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_5747_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_5748_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_5749_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_5750_unit__imp__mod__eq__0,axiom,
    ! [B: code_natural,A: code_natural] :
      ( ( dvd_dvd_Code_natural @ B @ one_one_Code_natural )
     => ( ( modulo8411746178871703098atural @ A @ B )
        = zero_z2226904508553997617atural ) ) ).

% unit_imp_mod_eq_0
thf(fact_5751_pochhammer__nonneg,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5752_pochhammer__nonneg,axiom,
    ! [X: rat,N: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5753_pochhammer__nonneg,axiom,
    ! [X: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5754_pochhammer__nonneg,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_int @ zero_zero_int @ X )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X @ N ) ) ) ).

% pochhammer_nonneg
thf(fact_5755_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_5756_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_5757_nat__mult__dvd__cancel1,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) )
        = ( dvd_dvd_nat @ M2 @ N ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_5758_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = one_one_Code_integer ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s8582702949713902594nteger @ zero_z3403309356797280102nteger @ N )
          = zero_z3403309356797280102nteger ) ) ) ).

% pochhammer_0_left
thf(fact_5759_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = one_one_complex ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N )
          = zero_zero_complex ) ) ) ).

% pochhammer_0_left
thf(fact_5760_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = one_one_real ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N )
          = zero_zero_real ) ) ) ).

% pochhammer_0_left
thf(fact_5761_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = one_one_rat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_5762_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = one_one_nat ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N )
          = zero_zero_nat ) ) ) ).

% pochhammer_0_left
thf(fact_5763_pochhammer__0__left,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = one_one_int ) )
      & ( ( N != zero_zero_nat )
       => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N )
          = zero_zero_int ) ) ) ).

% pochhammer_0_left
thf(fact_5764_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D5: nat,X3: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D5 @ A )
          & ( dvd_dvd_nat @ D5 @ B )
          & ( ( times_times_nat @ A @ X3 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D5 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_5765_mod__greater__zero__iff__not__dvd,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M2 @ N ) )
      = ( ~ ( dvd_dvd_nat @ N @ M2 ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_5766_mod__eq__dvd__iff__nat,axiom,
    ! [N: nat,M2: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( ( modulo_modulo_nat @ M2 @ Q2 )
          = ( modulo_modulo_nat @ N @ Q2 ) )
        = ( dvd_dvd_nat @ Q2 @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_5767_sum__power__add,axiom,
    ! [X: complex,M2: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I: nat] : ( power_power_complex @ X @ ( plus_plus_nat @ M2 @ I ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_5768_sum__power__add,axiom,
    ! [X: rat,M2: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( power_power_rat @ X @ ( plus_plus_nat @ M2 @ I ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_5769_sum__power__add,axiom,
    ! [X: int,M2: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( power_power_int @ X @ ( plus_plus_nat @ M2 @ I ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_5770_sum__power__add,axiom,
    ! [X: real,M2: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( power_power_real @ X @ ( plus_plus_nat @ M2 @ I ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_5771_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ ( groups8097168146408367636l_real @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5772_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > real] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ ( groups8778361861064173332t_real @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5773_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ ( groups5808333547571424918x_real @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5774_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ ( groups1300246762558778688al_rat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5775_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ ( groups3906332499630173760nt_rat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5776_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ ( groups5058264527183730370ex_rat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5777_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) @ ( groups1935376822645274424al_nat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5778_sum__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( groups4541462559716669496nt_nat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5779_sum__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( groups5693394587270226106ex_nat @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5780_sum__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F2 @ B3 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F2 @ A4 ) @ ( groups1932886352136224148al_int @ F2 @ B5 ) ) ) ) ) ).

% sum_mono2
thf(fact_5781_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_complex @ ( groups3049146728041665814omplex @ G2 @ A4 ) @ ( groups3049146728041665814omplex @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5782_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups7754918857620584856omplex @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G2 @ A4 ) @ ( groups7754918857620584856omplex @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5783_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G2 @ A4 ) @ ( groups8778361861064173332t_real @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5784_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G2 @ A4 ) @ ( groups5808333547571424918x_real @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5785_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G2 @ A4 ) @ ( groups3906332499630173760nt_rat @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5786_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G2 @ A4 ) @ ( groups5058264527183730370ex_rat @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5787_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups4541462559716669496nt_nat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G2 @ A4 ) @ ( groups4541462559716669496nt_nat @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5788_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G2 @ A4 ) @ ( groups5693394587270226106ex_nat @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5789_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G2 @ A4 ) @ ( groups5690904116761175830ex_int @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5790_sum_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G2 @ A4 ) @ ( groups2073611262835488442omplex @ G2 @ B5 ) ) ) ) ) ) ).

% sum.union_inter_neutral
thf(fact_5791_sum_OatLeastAtMost__rev,axiom,
    ! [G2: nat > nat,N: nat,M2: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5792_sum_OatLeastAtMost__rev,axiom,
    ! [G2: nat > real,N: nat,M2: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_5793_finite__divisors__nat,axiom,
    ! [M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M2 ) ) ) ) ).

% finite_divisors_nat
thf(fact_5794_powser__sums__zero,axiom,
    ! [A: nat > complex] :
      ( sums_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( A @ N4 ) @ ( power_power_complex @ zero_zero_complex @ N4 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_5795_powser__sums__zero,axiom,
    ! [A: nat > real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( A @ N4 ) @ ( power_power_real @ zero_zero_real @ N4 ) )
      @ ( A @ zero_zero_nat ) ) ).

% powser_sums_zero
thf(fact_5796_sum__shift__lb__Suc0__0,axiom,
    ! [F2: nat > complex,K2: nat] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( groups2073611262835488442omplex @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups2073611262835488442omplex @ F2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5797_sum__shift__lb__Suc0__0,axiom,
    ! [F2: nat > rat,K2: nat] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups2906978787729119204at_rat @ F2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5798_sum__shift__lb__Suc0__0,axiom,
    ! [F2: nat > int,K2: nat] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_int )
     => ( ( groups3539618377306564664at_int @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3539618377306564664at_int @ F2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5799_sum__shift__lb__Suc0__0,axiom,
    ! [F2: nat > nat,K2: nat] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_nat )
     => ( ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5800_sum__shift__lb__Suc0__0,axiom,
    ! [F2: nat > real,K2: nat] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_real )
     => ( ( groups6591440286371151544t_real @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K2 ) )
        = ( groups6591440286371151544t_real @ F2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_5801_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5802_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5803_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5804_sum_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_5805_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_rat @ ( G2 @ M2 ) @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5806_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_int @ ( G2 @ M2 ) @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5807_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_nat @ ( G2 @ M2 ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5808_sum_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( plus_plus_real @ ( G2 @ M2 ) @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_5809_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G2 @ ( suc @ N ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5810_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G2 @ ( suc @ N ) ) @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5811_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G2 @ ( suc @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5812_sum_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G2 @ ( suc @ N ) ) @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_5813_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_5814_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_5815_is__unitE,axiom,
    ! [A: code_integer,C2: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B3: code_integer] :
              ( ( B3 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B3 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B3 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B3 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B3 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C2 @ A )
                       != ( times_3573771949741848930nteger @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5816_is__unitE,axiom,
    ! [A: nat,C2: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B3: nat] :
              ( ( B3 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B3 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B3 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B3 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B3 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C2 @ A )
                       != ( times_times_nat @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5817_is__unitE,axiom,
    ! [A: int,C2: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B3: int] :
              ( ( B3 != zero_zero_int )
             => ( ( dvd_dvd_int @ B3 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B3 )
                 => ( ( ( divide_divide_int @ one_one_int @ B3 )
                      = A )
                   => ( ( ( times_times_int @ A @ B3 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C2 @ A )
                       != ( times_times_int @ C2 @ B3 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_5818_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5819_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5820_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_5821_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5822_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5823_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_5824_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_5825_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_5826_dvd__power__iff,axiom,
    ! [X: code_integer,M2: nat,N: nat] :
      ( ( X != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ N ) )
        = ( ( dvd_dvd_Code_integer @ X @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5827_dvd__power__iff,axiom,
    ! [X: nat,M2: nat,N: nat] :
      ( ( X != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X @ M2 ) @ ( power_power_nat @ X @ N ) )
        = ( ( dvd_dvd_nat @ X @ one_one_nat )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5828_dvd__power__iff,axiom,
    ! [X: int,M2: nat,N: nat] :
      ( ( X != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ N ) )
        = ( ( dvd_dvd_int @ X @ one_one_int )
          | ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).

% dvd_power_iff
thf(fact_5829_dvd__power,axiom,
    ! [N: nat,X: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X @ ( power_8256067586552552935nteger @ X @ N ) ) ) ).

% dvd_power
thf(fact_5830_dvd__power,axiom,
    ! [N: nat,X: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_nat ) )
     => ( dvd_dvd_nat @ X @ ( power_power_nat @ X @ N ) ) ) ).

% dvd_power
thf(fact_5831_dvd__power,axiom,
    ! [N: nat,X: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_real ) )
     => ( dvd_dvd_real @ X @ ( power_power_real @ X @ N ) ) ) ).

% dvd_power
thf(fact_5832_dvd__power,axiom,
    ! [N: nat,X: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_int ) )
     => ( dvd_dvd_int @ X @ ( power_power_int @ X @ N ) ) ) ).

% dvd_power
thf(fact_5833_dvd__power,axiom,
    ! [N: nat,X: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N )
        | ( X = one_one_complex ) )
     => ( dvd_dvd_complex @ X @ ( power_power_complex @ X @ N ) ) ) ).

% dvd_power
thf(fact_5834_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F2 @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ ( groups8097168146408367636l_real @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5835_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: int > real] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F2 @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
             => ( ord_less_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ ( groups8778361861064173332t_real @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5836_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F2 @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ ( groups5808333547571424918x_real @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5837_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ ( groups1300246762558778688al_rat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5838_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
             => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ ( groups3906332499630173760nt_rat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5839_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F2 @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ ( groups5058264527183730370ex_rat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5840_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: real > nat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) @ ( groups1935376822645274424al_nat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5841_sum__strict__mono2,axiom,
    ! [B5: set_int,A4: set_int,B: int,F2: int > nat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ( member_int @ B @ ( minus_minus_set_int @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
           => ( ! [X3: int] :
                  ( ( member_int @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
             => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( groups4541462559716669496nt_nat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5842_sum__strict__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,B: complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F2 @ B ) )
           => ( ! [X3: complex] :
                  ( ( member_complex @ X3 @ B5 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( groups5693394587270226106ex_nat @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5843_sum__strict__mono2,axiom,
    ! [B5: set_real,A4: set_real,B: real,F2: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B5 @ A4 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F2 @ B ) )
           => ( ! [X3: real] :
                  ( ( member_real @ X3 @ B5 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F2 @ A4 ) @ ( groups1932886352136224148al_int @ F2 @ B5 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_5844_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: complex > real] :
      ( ( member_complex @ I2 @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I2 ) @ ( groups5808333547571424918x_real @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5845_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: int > real] :
      ( ( member_int @ I2 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I2 ) @ ( groups8778361861064173332t_real @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5846_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: real > real] :
      ( ( member_real @ I2 @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I2 ) @ ( groups8097168146408367636l_real @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5847_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: complex > rat] :
      ( ( member_complex @ I2 @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5848_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: int > rat] :
      ( ( member_int @ I2 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5849_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: real > rat] :
      ( ( member_real @ I2 @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( groups1300246762558778688al_rat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5850_member__le__sum,axiom,
    ! [I2: nat,A4: set_nat,F2: nat > rat] :
      ( ( member_nat @ I2 @ A4 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ I2 @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_nat @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I2 ) @ ( groups2906978787729119204at_rat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5851_member__le__sum,axiom,
    ! [I2: complex,A4: set_complex,F2: complex > nat] :
      ( ( member_complex @ I2 @ A4 )
     => ( ! [X3: complex] :
            ( ( member_complex @ X3 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ I2 @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( finite3207457112153483333omplex @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5852_member__le__sum,axiom,
    ! [I2: int,A4: set_int,F2: int > nat] :
      ( ( member_int @ I2 @ A4 )
     => ( ! [X3: int] :
            ( ( member_int @ X3 @ ( minus_minus_set_int @ A4 @ ( insert_int @ I2 @ bot_bot_set_int ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_int @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5853_member__le__sum,axiom,
    ! [I2: real,A4: set_real,F2: real > nat] :
      ( ( member_real @ I2 @ A4 )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( minus_minus_set_real @ A4 @ ( insert_real @ I2 @ bot_bot_set_real ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
       => ( ( finite_finite_real @ A4 )
         => ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( groups1935376822645274424al_nat @ F2 @ A4 ) ) ) ) ) ).

% member_le_sum
thf(fact_5854_div2__even__ext__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_5855_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: real > rat,K5: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
       => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5856_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F2: nat > rat,K5: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
       => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5857_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: complex > rat,K5: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
       => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5858_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F2: int > rat,K5: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_rat @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
       => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5859_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: real > real,K5: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
       => ( ord_less_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5860_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: complex > real,K5: real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
       => ( ord_less_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5861_sum__bounded__above__strict,axiom,
    ! [A4: set_int,F2: int > real,K5: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_real @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_int @ A4 ) )
       => ( ord_less_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5862_sum__bounded__above__strict,axiom,
    ! [A4: set_real,F2: real > int,K5: int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_int @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_real @ A4 ) )
       => ( ord_less_int @ ( groups1932886352136224148al_int @ F2 @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_real @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5863_sum__bounded__above__strict,axiom,
    ! [A4: set_nat,F2: nat > int,K5: int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_int @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_nat @ A4 ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F2 @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_nat @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5864_sum__bounded__above__strict,axiom,
    ! [A4: set_complex,F2: complex > int,K5: int] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_int @ ( F2 @ I3 ) @ K5 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ ( finite_card_complex @ A4 ) )
       => ( ord_less_int @ ( groups5690904116761175830ex_int @ F2 @ A4 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( finite_card_complex @ A4 ) ) @ K5 ) ) ) ) ).

% sum_bounded_above_strict
thf(fact_5865_sum__bounded__above__divide,axiom,
    ! [A4: set_complex,F2: complex > real,K5: real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ A4 ) ) ) ) )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5866_sum__bounded__above__divide,axiom,
    ! [A4: set_int,F2: int > real,K5: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_int @ A4 ) ) ) ) )
     => ( ( finite_finite_int @ A4 )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5867_sum__bounded__above__divide,axiom,
    ! [A4: set_real,F2: real > real,K5: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_real @ A4 ) ) ) ) )
     => ( ( finite_finite_real @ A4 )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5868_sum__bounded__above__divide,axiom,
    ! [A4: set_complex,F2: complex > rat,K5: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( divide_divide_rat @ K5 @ ( semiri681578069525770553at_rat @ ( finite_card_complex @ A4 ) ) ) ) )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5869_sum__bounded__above__divide,axiom,
    ! [A4: set_nat,F2: nat > rat,K5: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( divide_divide_rat @ K5 @ ( semiri681578069525770553at_rat @ ( finite_card_nat @ A4 ) ) ) ) )
     => ( ( finite_finite_nat @ A4 )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5870_sum__bounded__above__divide,axiom,
    ! [A4: set_int,F2: int > rat,K5: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( divide_divide_rat @ K5 @ ( semiri681578069525770553at_rat @ ( finite_card_int @ A4 ) ) ) ) )
     => ( ( finite_finite_int @ A4 )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5871_sum__bounded__above__divide,axiom,
    ! [A4: set_real,F2: real > rat,K5: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( divide_divide_rat @ K5 @ ( semiri681578069525770553at_rat @ ( finite_card_real @ A4 ) ) ) ) )
     => ( ( finite_finite_real @ A4 )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5872_sum__bounded__above__divide,axiom,
    ! [A4: set_nat,F2: nat > real,K5: real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ A4 ) ) ) ) )
     => ( ( finite_finite_nat @ A4 )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5873_sum__bounded__above__divide,axiom,
    ! [A4: set_list_nat,F2: list_nat > real,K5: real] :
      ( ! [I3: list_nat] :
          ( ( member_list_nat @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ A4 ) ) ) ) )
     => ( ( finite8100373058378681591st_nat @ A4 )
       => ( ( A4 != bot_bot_set_list_nat )
         => ( ord_less_eq_real @ ( groups8399112307953289288t_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5874_sum__bounded__above__divide,axiom,
    ! [A4: set_set_nat,F2: set_nat > real,K5: real] :
      ( ! [I3: set_nat] :
          ( ( member_set_nat @ I3 @ A4 )
         => ( ord_less_eq_real @ ( F2 @ I3 ) @ ( divide_divide_real @ K5 @ ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ A4 ) ) ) ) )
     => ( ( finite1152437895449049373et_nat @ A4 )
       => ( ( A4 != bot_bot_set_set_nat )
         => ( ord_less_eq_real @ ( groups5107569545109728110t_real @ F2 @ A4 ) @ K5 ) ) ) ) ).

% sum_bounded_above_divide
thf(fact_5875_pochhammer__rec,axiom,
    ! [A: code_integer,N: nat] :
      ( ( comm_s8582702949713902594nteger @ A @ ( suc @ N ) )
      = ( times_3573771949741848930nteger @ A @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ N ) ) ) ).

% pochhammer_rec
thf(fact_5876_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_5877_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_5878_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_5879_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_5880_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_5881_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_5882_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_5883_pochhammer__rec_H,axiom,
    ! [Z3: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z3 @ ( suc @ N ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ N ) ) @ ( comm_s4028243227959126397er_rat @ Z3 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5884_pochhammer__rec_H,axiom,
    ! [Z3: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ Z3 @ ( suc @ N ) )
      = ( times_times_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ N ) ) @ ( comm_s7457072308508201937r_real @ Z3 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5885_pochhammer__rec_H,axiom,
    ! [Z3: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( suc @ N ) )
      = ( times_times_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N ) ) @ ( comm_s4660882817536571857er_int @ Z3 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5886_pochhammer__rec_H,axiom,
    ! [Z3: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( suc @ N ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N ) ) @ ( comm_s4663373288045622133er_nat @ Z3 @ N ) ) ) ).

% pochhammer_rec'
thf(fact_5887_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_5888_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_5889_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_5890_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_5891_dvd__minus__add,axiom,
    ! [Q2: nat,N: nat,R3: nat,M2: nat] :
      ( ( ord_less_eq_nat @ Q2 @ N )
     => ( ( ord_less_eq_nat @ Q2 @ ( times_times_nat @ R3 @ M2 ) )
       => ( ( dvd_dvd_nat @ M2 @ ( minus_minus_nat @ N @ Q2 ) )
          = ( dvd_dvd_nat @ M2 @ ( plus_plus_nat @ N @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M2 ) @ Q2 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_5892_power__dvd__imp__le,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M2 ) @ ( power_power_nat @ I2 @ N ) )
     => ( ( ord_less_nat @ one_one_nat @ I2 )
       => ( ord_less_eq_nat @ M2 @ N ) ) ) ).

% power_dvd_imp_le
thf(fact_5893_pochhammer__product_H,axiom,
    ! [Z3: rat,N: nat,M2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z3 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ N ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5894_pochhammer__product_H,axiom,
    ! [Z3: real,N: nat,M2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z3 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ N ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5895_pochhammer__product_H,axiom,
    ! [Z3: int,N: nat,M2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z3 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ N ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5896_pochhammer__product_H,axiom,
    ! [Z3: nat,N: nat,M2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z3 @ ( plus_plus_nat @ N @ M2 ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ N ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ N ) ) @ M2 ) ) ) ).

% pochhammer_product'
thf(fact_5897_mod__nat__eqI,axiom,
    ! [R3: nat,N: nat,M2: nat] :
      ( ( ord_less_nat @ R3 @ N )
     => ( ( ord_less_eq_nat @ R3 @ M2 )
       => ( ( dvd_dvd_nat @ N @ ( minus_minus_nat @ M2 @ R3 ) )
         => ( ( modulo_modulo_nat @ M2 @ N )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_5898_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( plus_plus_rat @ ( G2 @ M2 )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5899_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( plus_plus_int @ ( G2 @ M2 )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5900_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( plus_plus_nat @ ( G2 @ M2 )
          @ ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5901_sum_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( plus_plus_real @ ( G2 @ M2 )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_5902_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_rat @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5903_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( minus_minus_int @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_int @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5904_sum__Suc__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( minus_minus_real @ ( F2 @ ( suc @ I ) ) @ ( F2 @ I ) )
          @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( minus_minus_real @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M2 ) ) ) ) ).

% sum_Suc_diff
thf(fact_5905_sum__norm__bound,axiom,
    ! [S3: set_real,F2: real > complex,K5: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_real @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5906_sum__norm__bound,axiom,
    ! [S3: set_nat,F2: nat > complex,K5: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5907_sum__norm__bound,axiom,
    ! [S3: set_complex,F2: complex > complex,K5: real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_complex @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5908_sum__norm__bound,axiom,
    ! [S3: set_int,F2: int > complex,K5: real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_int @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5909_sum__norm__bound,axiom,
    ! [S3: set_list_nat,F2: list_nat > complex,K5: real] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6529277132148336714omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_list_nat @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5910_sum__norm__bound,axiom,
    ! [S3: set_set_nat,F2: set_nat > complex,K5: real] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups8255218700646806128omplex @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_set_nat @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5911_sum__norm__bound,axiom,
    ! [S3: set_nat,F2: nat > real,K5: real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ S3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F2 @ X3 ) ) @ K5 ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F2 @ S3 ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( finite_card_nat @ S3 ) ) @ K5 ) ) ) ).

% sum_norm_bound
thf(fact_5912_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > rat,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5913_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > int,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5914_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > nat,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5915_sum_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > real,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_5916_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_5917_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_5918_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_5919_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_5920_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_natural] :
      ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
      = ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
        = zero_z2226904508553997617atural ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_5921_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_5922_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_5923_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_5924_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_5925_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_5926_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_5927_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_5928_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_5929_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_5930_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_5931_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_5932_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_5933_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z3: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4028243227959126397er_rat @ Z3 @ N )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ M2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5934_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z3: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s7457072308508201937r_real @ Z3 @ N )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ M2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5935_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z3: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4660882817536571857er_int @ Z3 @ N )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z3 @ M2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z3 @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5936_pochhammer__product,axiom,
    ! [M2: nat,N: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( comm_s4663373288045622133er_nat @ Z3 @ N )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z3 @ M2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z3 @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ).

% pochhammer_product
thf(fact_5937_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_5938_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_5939_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_5940_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_5941_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_5942_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_5943_mod2__eq__if,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = zero_z2226904508553997617atural ) )
      & ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
          = one_one_Code_natural ) ) ) ).

% mod2_eq_if
thf(fact_5944_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_5945_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_5946_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_5947_parity__cases,axiom,
    ! [A: code_natural] :
      ( ( ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
       => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
         != zero_z2226904508553997617atural ) )
     => ~ ( ~ ( dvd_dvd_Code_natural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ A )
         => ( ( modulo8411746178871703098atural @ A @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) )
           != one_one_Code_natural ) ) ) ).

% parity_cases
thf(fact_5948_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_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > complex] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_complex @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_5958_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > rat] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_rat @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_5959_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > int] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_int @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_5960_sum__natinterval__diff,axiom,
    ! [M2: nat,N: nat,F2: nat > real] :
      ( ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = ( minus_minus_real @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_5961_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_rat @ ( F2 @ N ) @ ( F2 @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5962_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_int @ ( F2 @ N ) @ ( F2 @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5963_sum__telescope_H_H,axiom,
    ! [M2: nat,N: nat,F2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) )
        = ( minus_minus_real @ ( F2 @ N ) @ ( F2 @ M2 ) ) ) ) ).

% sum_telescope''
thf(fact_5964_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_5965_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_5966_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_5967_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_5968_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: code_integer] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ M2 ) @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5969_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X @ M2 ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5970_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X @ M2 ) @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5971_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_int @ ( power_power_int @ X @ M2 ) @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5972_sum__gp__multiplied,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) )
        = ( minus_minus_real @ ( power_power_real @ X @ M2 ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_5973_sum_Oin__pairs,axiom,
    ! [G2: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5974_sum_Oin__pairs,axiom,
    ! [G2: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( plus_plus_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5975_sum_Oin__pairs,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5976_sum_Oin__pairs,axiom,
    ! [G2: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( plus_plus_real @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% sum.in_pairs
thf(fact_5977_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_5978_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_5979_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_5980_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_5981_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_5982_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_5983_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
          @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N ) ) ) ) ).

% mask_eq_sum_exp_nat
thf(fact_5984_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_5985_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_5986_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_5987_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_5988_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_5989_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum
thf(fact_5990_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_5991_double__gauss__sum,axiom,
    ! [N: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum
thf(fact_5992_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_5993_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_5994_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_5995_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_5996_double__arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) )
        @ ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5997_double__arith__series,axiom,
    ! [A: rat,D: rat,N: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I ) @ 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_5998_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_5999_double__arith__series,axiom,
    ! [A: complex,D: complex,N: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I ) @ 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_6000_double__arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I ) @ 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_6001_double__arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I ) @ 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_6002_double__arith__series,axiom,
    ! [A: real,D: real,N: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I ) @ 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_6003_arith__series__nat,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_6004_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_6005_Bernoulli__inequality__even,axiom,
    ! [N: nat,X: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality_even
thf(fact_6006_gauss__sum,axiom,
    ! [N: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_6007_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_6008_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_6009_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_6010_arith__series,axiom,
    ! [A: int,D: int,N: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I ) @ 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_6011_arith__series,axiom,
    ! [A: nat,D: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I ) @ 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_6012_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6013_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_6014_double__gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) ) )
      = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N ) @ one_on7984719198319812577d_enat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_6015_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_6016_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_6017_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_6018_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_6019_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_6020_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_6021_sum__gp__offset,axiom,
    ! [X: complex,M2: nat,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N ) @ one_one_complex ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6022_sum__gp__offset,axiom,
    ! [X: rat,M2: nat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N ) @ one_one_rat ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6023_sum__gp__offset,axiom,
    ! [X: real,M2: nat,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ M2 @ N ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_offset
thf(fact_6024_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va3: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va3 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
          = ( 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 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
          = ( 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.simps(3)
thf(fact_6025_gauss__sum__from__Suc__0,axiom,
    ! [N: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_6026_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_6027_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_6028_power__half__series,axiom,
    ( sums_real
    @ ^ [N4: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N4 ) )
    @ one_one_real ) ).

% power_half_series
thf(fact_6029_sums__zero,axiom,
    ( sums_complex
    @ ^ [N4: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_6030_sums__zero,axiom,
    ( sums_real
    @ ^ [N4: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_6031_sums__zero,axiom,
    ( sums_nat
    @ ^ [N4: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_6032_sums__zero,axiom,
    ( sums_int
    @ ^ [N4: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_6033_powser__sums__if,axiom,
    ! [M2: nat,Z3: complex] :
      ( sums_complex
      @ ^ [N4: nat] : ( times_times_complex @ ( if_complex @ ( N4 = M2 ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z3 @ N4 ) )
      @ ( power_power_complex @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6034_powser__sums__if,axiom,
    ! [M2: nat,Z3: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( if_real @ ( N4 = M2 ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z3 @ N4 ) )
      @ ( power_power_real @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6035_powser__sums__if,axiom,
    ! [M2: nat,Z3: int] :
      ( sums_int
      @ ^ [N4: nat] : ( times_times_int @ ( if_int @ ( N4 = M2 ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z3 @ N4 ) )
      @ ( power_power_int @ Z3 @ M2 ) ) ).

% powser_sums_if
thf(fact_6036_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( sums_complex
        @ ^ [R: nat] : ( if_complex @ ( member_nat @ R @ A4 ) @ ( F2 @ R ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F2 @ A4 ) ) ) ).

% sums_If_finite_set
thf(fact_6037_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( sums_int
        @ ^ [R: nat] : ( if_int @ ( member_nat @ R @ A4 ) @ ( F2 @ R ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F2 @ A4 ) ) ) ).

% sums_If_finite_set
thf(fact_6038_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( sums_nat
        @ ^ [R: nat] : ( if_nat @ ( member_nat @ R @ A4 ) @ ( F2 @ R ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F2 @ A4 ) ) ) ).

% sums_If_finite_set
thf(fact_6039_sums__If__finite__set,axiom,
    ! [A4: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( sums_real
        @ ^ [R: nat] : ( if_real @ ( member_nat @ R @ A4 ) @ ( F2 @ R ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F2 @ A4 ) ) ) ).

% sums_If_finite_set
thf(fact_6040_sums__If__finite,axiom,
    ! [P2: nat > $o,F2: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_complex
        @ ^ [R: nat] : ( if_complex @ ( P2 @ R ) @ ( F2 @ R ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F2 @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6041_sums__If__finite,axiom,
    ! [P2: nat > $o,F2: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_int
        @ ^ [R: nat] : ( if_int @ ( P2 @ R ) @ ( F2 @ R ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F2 @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6042_sums__If__finite,axiom,
    ! [P2: nat > $o,F2: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_nat
        @ ^ [R: nat] : ( if_nat @ ( P2 @ R ) @ ( F2 @ R ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F2 @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6043_sums__If__finite,axiom,
    ! [P2: nat > $o,F2: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P2 ) )
     => ( sums_real
        @ ^ [R: nat] : ( if_real @ ( P2 @ R ) @ ( F2 @ R ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F2 @ ( collect_nat @ P2 ) ) ) ) ).

% sums_If_finite
thf(fact_6044_sums__finite,axiom,
    ! [N6: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F2 @ N3 )
              = zero_zero_complex ) )
       => ( sums_complex @ F2 @ ( groups2073611262835488442omplex @ F2 @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6045_sums__finite,axiom,
    ! [N6: set_nat,F2: nat > int] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F2 @ N3 )
              = zero_zero_int ) )
       => ( sums_int @ F2 @ ( groups3539618377306564664at_int @ F2 @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6046_sums__finite,axiom,
    ! [N6: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F2 @ N3 )
              = zero_zero_nat ) )
       => ( sums_nat @ F2 @ ( groups3542108847815614940at_nat @ F2 @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6047_sums__finite,axiom,
    ! [N6: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ N6 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N6 )
           => ( ( F2 @ N3 )
              = zero_zero_real ) )
       => ( sums_real @ F2 @ ( groups6591440286371151544t_real @ F2 @ N6 ) ) ) ) ).

% sums_finite
thf(fact_6048_sums__zero__iff__shift,axiom,
    ! [N: nat,F2: nat > complex,S2: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F2 @ I3 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I: nat] : ( F2 @ ( plus_plus_nat @ I @ N ) )
          @ S2 )
        = ( sums_complex @ F2 @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_6049_sums__zero__iff__shift,axiom,
    ! [N: nat,F2: nat > real,S2: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ( F2 @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I: nat] : ( F2 @ ( plus_plus_nat @ I @ N ) )
          @ S2 )
        = ( sums_real @ F2 @ S2 ) ) ) ).

% sums_zero_iff_shift
thf(fact_6050_zdvd__mono,axiom,
    ! [K2: int,M2: int,T: int] :
      ( ( K2 != zero_zero_int )
     => ( ( dvd_dvd_int @ M2 @ T )
        = ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ T ) ) ) ) ).

% zdvd_mono
thf(fact_6051_zdvd__mult__cancel,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) )
     => ( ( K2 != zero_zero_int )
       => ( dvd_dvd_int @ M2 @ N ) ) ) ).

% zdvd_mult_cancel
thf(fact_6052_zdvd__period,axiom,
    ! [A: int,D: int,X: int,T: int,C2: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X @ ( times_times_int @ C2 @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_6053_zdvd__reduce,axiom,
    ! [K2: int,N: int,M2: int] :
      ( ( dvd_dvd_int @ K2 @ ( plus_plus_int @ N @ ( times_times_int @ K2 @ M2 ) ) )
      = ( dvd_dvd_int @ K2 @ N ) ) ).

% zdvd_reduce
thf(fact_6054_sum__subtractf__nat,axiom,
    ! [A4: set_complex,G2: complex > nat,F2: complex > nat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( minus_minus_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( groups5693394587270226106ex_nat @ G2 @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6055_sum__subtractf__nat,axiom,
    ! [A4: set_real,G2: real > nat,F2: real > nat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( minus_minus_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F2 @ A4 ) @ ( groups1935376822645274424al_nat @ G2 @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6056_sum__subtractf__nat,axiom,
    ! [A4: set_set_nat,G2: set_nat > nat,F2: set_nat > nat] :
      ( ! [X3: set_nat] :
          ( ( member_set_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X4: set_nat] : ( minus_minus_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F2 @ A4 ) @ ( groups8294997508430121362at_nat @ G2 @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6057_sum__subtractf__nat,axiom,
    ! [A4: set_int,G2: int > nat,F2: int > nat] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( minus_minus_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( groups4541462559716669496nt_nat @ G2 @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6058_sum__subtractf__nat,axiom,
    ! [A4: set_nat,G2: nat > nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ ( G2 @ X3 ) @ ( F2 @ X3 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( F2 @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F2 @ A4 ) @ ( groups3542108847815614940at_nat @ G2 @ A4 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_6059_sum__eq__Suc0__iff,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups4541462559716669496nt_nat @ F2 @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6060_sum__eq__Suc0__iff,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F2 @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6061_sum__eq__Suc0__iff,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F2 @ A4 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = ( suc @ zero_zero_nat ) )
              & ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_6062_sum__SucD,axiom,
    ! [F2: nat > nat,A4: set_nat,N: nat] :
      ( ( ( groups3542108847815614940at_nat @ F2 @ A4 )
        = ( suc @ N ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( ord_less_nat @ zero_zero_nat @ ( F2 @ X3 ) ) ) ) ).

% sum_SucD
thf(fact_6063_sum__eq__1__iff,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups4541462559716669496nt_nat @ F2 @ A4 )
          = one_one_nat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = one_one_nat )
              & ! [Y4: int] :
                  ( ( member_int @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6064_sum__eq__1__iff,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups5693394587270226106ex_nat @ F2 @ A4 )
          = one_one_nat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = one_one_nat )
              & ! [Y4: complex] :
                  ( ( member_complex @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6065_sum__eq__1__iff,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups3542108847815614940at_nat @ F2 @ A4 )
          = one_one_nat )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = one_one_nat )
              & ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ A4 )
                 => ( ( X4 != Y4 )
                   => ( ( F2 @ Y4 )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_6066_sum__Suc,axiom,
    ! [F2: complex > nat,A4: set_complex] :
      ( ( groups5693394587270226106ex_nat
        @ ^ [X4: complex] : ( suc @ ( F2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ F2 @ A4 ) @ ( finite_card_complex @ A4 ) ) ) ).

% sum_Suc
thf(fact_6067_sum__Suc,axiom,
    ! [F2: int > nat,A4: set_int] :
      ( ( groups4541462559716669496nt_nat
        @ ^ [X4: int] : ( suc @ ( F2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ F2 @ A4 ) @ ( finite_card_int @ A4 ) ) ) ).

% sum_Suc
thf(fact_6068_sum__Suc,axiom,
    ! [F2: list_nat > nat,A4: set_list_nat] :
      ( ( groups4396056296759096172at_nat
        @ ^ [X4: list_nat] : ( suc @ ( F2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups4396056296759096172at_nat @ F2 @ A4 ) @ ( finite_card_list_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6069_sum__Suc,axiom,
    ! [F2: set_nat > nat,A4: set_set_nat] :
      ( ( groups8294997508430121362at_nat
        @ ^ [X4: set_nat] : ( suc @ ( F2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups8294997508430121362at_nat @ F2 @ A4 ) @ ( finite_card_set_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6070_sum__Suc,axiom,
    ! [F2: nat > nat,A4: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X4: nat] : ( suc @ ( F2 @ X4 ) )
        @ A4 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F2 @ A4 ) @ ( finite_card_nat @ A4 ) ) ) ).

% sum_Suc
thf(fact_6071_sum__multicount,axiom,
    ! [S3: set_real,T5: set_real,R2: real > real > $o,K2: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T5 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I: real] :
                        ( ( member_real @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I: real] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J: real] :
                        ( ( member_real @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_real @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6072_sum__multicount,axiom,
    ! [S3: set_real,T5: set_nat,R2: real > nat > $o,K2: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I: real] :
                        ( ( member_real @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I: real] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6073_sum__multicount,axiom,
    ! [S3: set_real,T5: set_int,R2: real > int > $o,K2: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I: real] :
                        ( ( member_real @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I: real] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_int @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6074_sum__multicount,axiom,
    ! [S3: set_real,T5: set_complex,R2: real > complex > $o,K2: nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T5 )
             => ( ( finite_card_real
                  @ ( collect_real
                    @ ^ [I: real] :
                        ( ( member_real @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups1935376822645274424al_nat
              @ ^ [I: real] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J: complex] :
                        ( ( member_complex @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_complex @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6075_sum__multicount,axiom,
    ! [S3: set_int,T5: set_real,R2: int > real > $o,K2: nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J: real] :
                        ( ( member_real @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_real @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6076_sum__multicount,axiom,
    ! [S3: set_int,T5: set_nat,R2: int > nat > $o,K2: nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6077_sum__multicount,axiom,
    ! [S3: set_int,T5: set_int,R2: int > int > $o,K2: nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_int
                  @ ( collect_int
                    @ ^ [J: int] :
                        ( ( member_int @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_int @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6078_sum__multicount,axiom,
    ! [S3: set_int,T5: set_complex,R2: int > complex > $o,K2: nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ T5 )
             => ( ( finite_card_int
                  @ ( collect_int
                    @ ^ [I: int] :
                        ( ( member_int @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups4541462559716669496nt_nat
              @ ^ [I: int] :
                  ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [J: complex] :
                        ( ( member_complex @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_complex @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6079_sum__multicount,axiom,
    ! [S3: set_complex,T5: set_real,R2: complex > real > $o,K2: nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [X3: real] :
              ( ( member_real @ X3 @ T5 )
             => ( ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [I: complex] :
                        ( ( member_complex @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [I: complex] :
                  ( finite_card_real
                  @ ( collect_real
                    @ ^ [J: real] :
                        ( ( member_real @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_real @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6080_sum__multicount,axiom,
    ! [S3: set_complex,T5: set_nat,R2: complex > nat > $o,K2: nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( finite_finite_nat @ T5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ T5 )
             => ( ( finite_card_complex
                  @ ( collect_complex
                    @ ^ [I: complex] :
                        ( ( member_complex @ I @ S3 )
                        & ( R2 @ I @ X3 ) ) ) )
                = K2 ) )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [I: complex] :
                  ( finite_card_nat
                  @ ( collect_nat
                    @ ^ [J: nat] :
                        ( ( member_nat @ J @ T5 )
                        & ( R2 @ I @ J ) ) ) )
              @ S3 )
            = ( times_times_nat @ K2 @ ( finite_card_nat @ T5 ) ) ) ) ) ) ).

% sum_multicount
thf(fact_6081_mod__int__pos__iff,axiom,
    ! [K2: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K2 @ L ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K2 ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_6082_sum__count__set,axiom,
    ! [Xs2: list_complex,X8: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ X8 )
     => ( ( finite3207457112153483333omplex @ X8 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs2 ) @ X8 )
          = ( size_s3451745648224563538omplex @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6083_sum__count__set,axiom,
    ! [Xs2: list_VEBT_VEBT,X8: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ X8 )
     => ( ( finite5795047828879050333T_VEBT @ X8 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs2 ) @ X8 )
          = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6084_sum__count__set,axiom,
    ! [Xs2: list_o,X8: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ X8 )
     => ( ( finite_finite_o @ X8 )
       => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs2 ) @ X8 )
          = ( size_size_list_o @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6085_sum__count__set,axiom,
    ! [Xs2: list_int,X8: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ X8 )
     => ( ( finite_finite_int @ X8 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs2 ) @ X8 )
          = ( size_size_list_int @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6086_sum__count__set,axiom,
    ! [Xs2: list_nat,X8: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ X8 )
     => ( ( finite_finite_nat @ X8 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs2 ) @ X8 )
          = ( size_size_list_nat @ Xs2 ) ) ) ) ).

% sum_count_set
thf(fact_6087_sums__le,axiom,
    ! [F2: nat > real,G2: nat > real,S2: real,T: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F2 @ N3 ) @ ( G2 @ N3 ) )
     => ( ( sums_real @ F2 @ S2 )
       => ( ( sums_real @ G2 @ T )
         => ( ord_less_eq_real @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6088_sums__le,axiom,
    ! [F2: nat > nat,G2: nat > nat,S2: nat,T: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F2 @ N3 ) @ ( G2 @ N3 ) )
     => ( ( sums_nat @ F2 @ S2 )
       => ( ( sums_nat @ G2 @ T )
         => ( ord_less_eq_nat @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6089_sums__le,axiom,
    ! [F2: nat > int,G2: nat > int,S2: int,T: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F2 @ N3 ) @ ( G2 @ N3 ) )
     => ( ( sums_int @ F2 @ S2 )
       => ( ( sums_int @ G2 @ T )
         => ( ord_less_eq_int @ S2 @ T ) ) ) ) ).

% sums_le
thf(fact_6090_sums__0,axiom,
    ! [F2: nat > complex] :
      ( ! [N3: nat] :
          ( ( F2 @ N3 )
          = zero_zero_complex )
     => ( sums_complex @ F2 @ zero_zero_complex ) ) ).

% sums_0
thf(fact_6091_sums__0,axiom,
    ! [F2: nat > real] :
      ( ! [N3: nat] :
          ( ( F2 @ N3 )
          = zero_zero_real )
     => ( sums_real @ F2 @ zero_zero_real ) ) ).

% sums_0
thf(fact_6092_sums__0,axiom,
    ! [F2: nat > nat] :
      ( ! [N3: nat] :
          ( ( F2 @ N3 )
          = zero_zero_nat )
     => ( sums_nat @ F2 @ zero_zero_nat ) ) ).

% sums_0
thf(fact_6093_sums__0,axiom,
    ! [F2: nat > int] :
      ( ! [N3: nat] :
          ( ( F2 @ N3 )
          = zero_zero_int )
     => ( sums_int @ F2 @ zero_zero_int ) ) ).

% sums_0
thf(fact_6094_sums__single,axiom,
    ! [I2: nat,F2: nat > complex] :
      ( sums_complex
      @ ^ [R: nat] : ( if_complex @ ( R = I2 ) @ ( F2 @ R ) @ zero_zero_complex )
      @ ( F2 @ I2 ) ) ).

% sums_single
thf(fact_6095_sums__single,axiom,
    ! [I2: nat,F2: nat > real] :
      ( sums_real
      @ ^ [R: nat] : ( if_real @ ( R = I2 ) @ ( F2 @ R ) @ zero_zero_real )
      @ ( F2 @ I2 ) ) ).

% sums_single
thf(fact_6096_sums__single,axiom,
    ! [I2: nat,F2: nat > nat] :
      ( sums_nat
      @ ^ [R: nat] : ( if_nat @ ( R = I2 ) @ ( F2 @ R ) @ zero_zero_nat )
      @ ( F2 @ I2 ) ) ).

% sums_single
thf(fact_6097_sums__single,axiom,
    ! [I2: nat,F2: nat > int] :
      ( sums_int
      @ ^ [R: nat] : ( if_int @ ( R = I2 ) @ ( F2 @ R ) @ zero_zero_int )
      @ ( F2 @ I2 ) ) ).

% sums_single
thf(fact_6098_sums__mult2,axiom,
    ! [F2: nat > real,A: real,C2: real] :
      ( ( sums_real @ F2 @ A )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ ( F2 @ N4 ) @ C2 )
        @ ( times_times_real @ A @ C2 ) ) ) ).

% sums_mult2
thf(fact_6099_sums__mult,axiom,
    ! [F2: nat > real,A: real,C2: real] :
      ( ( sums_real @ F2 @ A )
     => ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ C2 @ ( F2 @ N4 ) )
        @ ( times_times_real @ C2 @ A ) ) ) ).

% sums_mult
thf(fact_6100_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_6101_sums__mult__iff,axiom,
    ! [C2: complex,F2: nat > complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F2 @ N4 ) )
          @ ( times_times_complex @ C2 @ D ) )
        = ( sums_complex @ F2 @ D ) ) ) ).

% sums_mult_iff
thf(fact_6102_sums__mult__iff,axiom,
    ! [C2: real,F2: nat > real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( times_times_real @ C2 @ ( F2 @ N4 ) )
          @ ( times_times_real @ C2 @ D ) )
        = ( sums_real @ F2 @ D ) ) ) ).

% sums_mult_iff
thf(fact_6103_sums__mult2__iff,axiom,
    ! [C2: complex,F2: nat > complex,D: complex] :
      ( ( C2 != zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( times_times_complex @ ( F2 @ N4 ) @ C2 )
          @ ( times_times_complex @ D @ C2 ) )
        = ( sums_complex @ F2 @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6104_sums__mult2__iff,axiom,
    ! [C2: real,F2: nat > real,D: real] :
      ( ( C2 != zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( times_times_real @ ( F2 @ N4 ) @ C2 )
          @ ( times_times_real @ D @ C2 ) )
        = ( sums_real @ F2 @ D ) ) ) ).

% sums_mult2_iff
thf(fact_6105_sums__mult__D,axiom,
    ! [C2: complex,F2: nat > complex,A: complex] :
      ( ( sums_complex
        @ ^ [N4: nat] : ( times_times_complex @ C2 @ ( F2 @ N4 ) )
        @ A )
     => ( ( C2 != zero_zero_complex )
       => ( sums_complex @ F2 @ ( divide1717551699836669952omplex @ A @ C2 ) ) ) ) ).

% sums_mult_D
thf(fact_6106_sums__mult__D,axiom,
    ! [C2: real,F2: nat > real,A: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( times_times_real @ C2 @ ( F2 @ N4 ) )
        @ A )
     => ( ( C2 != zero_zero_real )
       => ( sums_real @ F2 @ ( divide_divide_real @ A @ C2 ) ) ) ) ).

% sums_mult_D
thf(fact_6107_sums__Suc__imp,axiom,
    ! [F2: nat > complex,S2: complex] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_complex )
     => ( ( sums_complex
          @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
          @ S2 )
       => ( sums_complex @ F2 @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_6108_sums__Suc__imp,axiom,
    ! [F2: nat > real,S2: real] :
      ( ( ( F2 @ zero_zero_nat )
        = zero_zero_real )
     => ( ( sums_real
          @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
          @ S2 )
       => ( sums_real @ F2 @ S2 ) ) ) ).

% sums_Suc_imp
thf(fact_6109_sums__Suc__iff,axiom,
    ! [F2: nat > real,S2: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
        @ S2 )
      = ( sums_real @ F2 @ ( plus_plus_real @ S2 @ ( F2 @ zero_zero_nat ) ) ) ) ).

% sums_Suc_iff
thf(fact_6110_sums__Suc,axiom,
    ! [F2: nat > real,L: real] :
      ( ( sums_real
        @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
        @ L )
     => ( sums_real @ F2 @ ( plus_plus_real @ L @ ( F2 @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6111_sums__Suc,axiom,
    ! [F2: nat > nat,L: nat] :
      ( ( sums_nat
        @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
        @ L )
     => ( sums_nat @ F2 @ ( plus_plus_nat @ L @ ( F2 @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6112_sums__Suc,axiom,
    ! [F2: nat > int,L: int] :
      ( ( sums_int
        @ ^ [N4: nat] : ( F2 @ ( suc @ N4 ) )
        @ L )
     => ( sums_int @ F2 @ ( plus_plus_int @ L @ ( F2 @ zero_zero_nat ) ) ) ) ).

% sums_Suc
thf(fact_6113_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: complex,N: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z3 @ ( suc @ N ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z3 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z3 @ ( 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_6114_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: rat,N: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z3 @ ( suc @ N ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z3 @ ( 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_6115_pochhammer__times__pochhammer__half,axiom,
    ! [Z3: real,N: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z3 @ ( suc @ N ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z3 @ ( 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_6116_lemma__termdiff2,axiom,
    ! [H: complex,Z3: complex,N: nat] :
      ( ( H != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H ) @ N ) @ ( power_power_complex @ Z3 @ N ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H ) @ Q5 ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( 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_6117_lemma__termdiff2,axiom,
    ! [H: rat,Z3: rat,N: nat] :
      ( ( H != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z3 @ H ) @ N ) @ ( power_power_rat @ Z3 @ N ) ) @ H ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( power_power_rat @ Z3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z3 @ H ) @ Q5 ) @ ( power_power_rat @ Z3 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( 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_6118_lemma__termdiff2,axiom,
    ! [H: real,Z3: real,N: nat] :
      ( ( H != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H ) @ N ) @ ( power_power_real @ Z3 @ N ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H ) @ Q5 ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ ( minus_minus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
                @ ( 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_6119_pochhammer__code,axiom,
    ( comm_s8582702949713902594nteger
    = ( ^ [A5: code_integer,N4: nat] :
          ( if_Code_integer @ ( N4 = zero_zero_nat ) @ one_one_Code_integer
          @ ( set_fo1084959871951514735nteger
            @ ^ [O: nat] : ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ A5 @ ( semiri4939895301339042750nteger @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_Code_integer ) ) ) ) ).

% pochhammer_code
thf(fact_6120_pochhammer__code,axiom,
    ( comm_s2602460028002588243omplex
    = ( ^ [A5: complex,N4: nat] :
          ( if_complex @ ( N4 = zero_zero_nat ) @ one_one_complex
          @ ( set_fo1517530859248394432omplex
            @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A5 @ ( semiri8010041392384452111omplex @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_complex ) ) ) ) ).

% pochhammer_code
thf(fact_6121_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A5: rat,N4: nat] :
          ( if_rat @ ( N4 = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A5 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_6122_pochhammer__code,axiom,
    ( comm_s7457072308508201937r_real
    = ( ^ [A5: real,N4: nat] :
          ( if_real @ ( N4 = zero_zero_nat ) @ one_one_real
          @ ( set_fo3111899725591712190t_real
            @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A5 @ ( semiri5074537144036343181t_real @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_real ) ) ) ) ).

% pochhammer_code
thf(fact_6123_pochhammer__code,axiom,
    ( comm_s4660882817536571857er_int
    = ( ^ [A5: int,N4: nat] :
          ( if_int @ ( N4 = zero_zero_nat ) @ one_one_int
          @ ( set_fo2581907887559384638at_int
            @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A5 @ ( semiri1314217659103216013at_int @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_int ) ) ) ) ).

% pochhammer_code
thf(fact_6124_pochhammer__code,axiom,
    ( comm_s4663373288045622133er_nat
    = ( ^ [A5: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ one_one_nat
          @ ( set_fo2584398358068434914at_nat
            @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A5 @ ( semiri1316708129612266289at_nat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N4 @ one_one_nat )
            @ one_one_nat ) ) ) ) ).

% pochhammer_code
thf(fact_6125_of__nat__code,axiom,
    ( semiri4939895301339042750nteger
    = ( ^ [N4: nat] :
          ( semiri4055485073559036834nteger
          @ ^ [I: code_integer] : ( plus_p5714425477246183910nteger @ I @ one_one_Code_integer )
          @ N4
          @ zero_z3403309356797280102nteger ) ) ) ).

% of_nat_code
thf(fact_6126_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N4: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I: complex] : ( plus_plus_complex @ I @ one_one_complex )
          @ N4
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_6127_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N4: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I: rat] : ( plus_plus_rat @ I @ one_one_rat )
          @ N4
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_6128_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N4: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I: real] : ( plus_plus_real @ I @ one_one_real )
          @ N4
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_6129_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I: int] : ( plus_plus_int @ I @ one_one_int )
          @ N4
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_6130_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I: nat] : ( plus_plus_nat @ I @ one_one_nat )
          @ N4
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_6131_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_6132_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_6133_dbl__simps_I3_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6134_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6135_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6136_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_6137_lessThan__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X = Y ) ) ).

% lessThan_eq_iff
thf(fact_6138_lessThan__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or890127255671739683et_nat @ K2 ) )
      = ( ord_less_set_nat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6139_lessThan__iff,axiom,
    ! [I2: real,K2: real] :
      ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K2 ) )
      = ( ord_less_real @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6140_lessThan__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K2 ) )
      = ( ord_less_rat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6141_lessThan__iff,axiom,
    ! [I2: num,K2: num] :
      ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K2 ) )
      = ( ord_less_num @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6142_lessThan__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K2 ) )
      = ( ord_less_int @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6143_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K2 ) )
      = ( ord_less_nat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_6144_finite__lessThan,axiom,
    ! [K2: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K2 ) ) ).

% finite_lessThan
thf(fact_6145_concat__bit__0,axiom,
    ! [K2: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K2 @ L )
      = L ) ).

% concat_bit_0
thf(fact_6146_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_6147_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_6148_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_6149_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_6150_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_6151_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups6464643781859351333omplex @ F2 @ A4 )
          = zero_zero_complex )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6152_prod__zero__iff,axiom,
    ! [A4: set_int,F2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups7440179247065528705omplex @ F2 @ A4 )
          = zero_zero_complex )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6153_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups3708469109370488835omplex @ F2 @ A4 )
          = zero_zero_complex )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6154_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups129246275422532515t_real @ F2 @ A4 )
          = zero_zero_real )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6155_prod__zero__iff,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups2316167850115554303t_real @ F2 @ A4 )
          = zero_zero_real )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6156_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups766887009212190081x_real @ F2 @ A4 )
          = zero_zero_real )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6157_prod__zero__iff,axiom,
    ! [A4: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( groups73079841787564623at_rat @ F2 @ A4 )
          = zero_zero_rat )
        = ( ? [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6158_prod__zero__iff,axiom,
    ! [A4: set_int,F2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups1072433553688619179nt_rat @ F2 @ A4 )
          = zero_zero_rat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6159_prod__zero__iff,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( groups225925009352817453ex_rat @ F2 @ A4 )
          = zero_zero_rat )
        = ( ? [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6160_prod__zero__iff,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( groups1707563613775114915nt_nat @ F2 @ A4 )
          = zero_zero_nat )
        = ( ? [X4: int] :
              ( ( member_int @ X4 @ A4 )
              & ( ( F2 @ X4 )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_6161_lessThan__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6162_lessThan__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6163_lessThan__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6164_lessThan__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% lessThan_subset_iff
thf(fact_6165_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_6166_dbl__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K2 ) ) ) ).

% dbl_simps(5)
thf(fact_6167_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_6168_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_6169_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A4 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6170_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A4 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6171_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A4 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6172_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A4 ) )
          = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6173_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A4 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6174_prod_Oinsert,axiom,
    ! [A4: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ~ ( member_nat @ X @ A4 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A4 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6175_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A4 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6176_prod_Oinsert,axiom,
    ! [A4: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ~ ( member_complex @ X @ A4 )
       => ( ( groups225925009352817453ex_rat @ G2 @ ( insert_complex @ X @ A4 ) )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6177_prod_Oinsert,axiom,
    ! [A4: set_real,X: real,G2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ~ ( member_real @ X @ A4 )
       => ( ( groups4696554848551431203al_nat @ G2 @ ( insert_real @ X @ A4 ) )
          = ( times_times_nat @ ( G2 @ X ) @ ( groups4696554848551431203al_nat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6178_prod_Oinsert,axiom,
    ! [A4: set_int,X: int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ~ ( member_int @ X @ A4 )
       => ( ( groups1707563613775114915nt_nat @ G2 @ ( insert_int @ X @ A4 ) )
          = ( times_times_nat @ ( G2 @ X ) @ ( groups1707563613775114915nt_nat @ G2 @ A4 ) ) ) ) ) ).

% prod.insert
thf(fact_6179_sum_OlessThan__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6180_sum_OlessThan__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6181_sum_OlessThan__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6182_sum_OlessThan__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% sum.lessThan_Suc
thf(fact_6183_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_6184_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_6185_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_6186_prod_OlessThan__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_6187_prod_OlessThan__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_6188_prod_OlessThan__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_6189_prod_OlessThan__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) @ ( G2 @ N ) ) ) ).

% prod.lessThan_Suc
thf(fact_6190_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > code_integer] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_Code_integer ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6191_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6192_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6193_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6194_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6195_prod_Ocl__ivl__Suc,axiom,
    ! [N: nat,M2: nat,G2: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N ) @ M2 )
       => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_6196_prod_Onat__diff__reindex,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6197_prod_Onat__diff__reindex,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nat_diff_reindex
thf(fact_6198_lessThan__non__empty,axiom,
    ! [X: int] :
      ( ( set_ord_lessThan_int @ X )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_6199_lessThan__non__empty,axiom,
    ! [X: real] :
      ( ( set_or5984915006950818249n_real @ X )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_6200_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_6201_prod_Odistrib,axiom,
    ! [G2: nat > int,H: nat > int,A4: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [X4: nat] : ( times_times_int @ ( G2 @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ A4 ) @ ( groups705719431365010083at_int @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_6202_prod_Odistrib,axiom,
    ! [G2: int > int,H: int > int,A4: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [X4: int] : ( times_times_int @ ( G2 @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_int @ ( groups1705073143266064639nt_int @ G2 @ A4 ) @ ( groups1705073143266064639nt_int @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_6203_prod_Odistrib,axiom,
    ! [G2: nat > nat,H: nat > nat,A4: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [X4: nat] : ( times_times_nat @ ( G2 @ X4 ) @ ( H @ X4 ) )
        @ A4 )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ A4 ) @ ( groups708209901874060359at_nat @ H @ A4 ) ) ) ).

% prod.distrib
thf(fact_6204_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X4: set_nat] : ( ord_less_set_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6205_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_real @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6206_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_rat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6207_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_num @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6208_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_int @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6209_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_nat @ X4 @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_6210_prod_OlessThan__Suc__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G2 @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6211_prod_OlessThan__Suc__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G2 @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6212_prod_OlessThan__Suc__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G2 @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6213_prod_OlessThan__Suc__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G2 @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.lessThan_Suc_shift
thf(fact_6214_prod_OatLeast1__atMost__eq,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6215_prod_OatLeast1__atMost__eq,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.atLeast1_atMost_eq
thf(fact_6216_prod__mono,axiom,
    ! [A4: set_complex,F2: complex > real,G2: complex > real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( groups766887009212190081x_real @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6217_prod__mono,axiom,
    ! [A4: set_real,F2: real > real,G2: real > real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ ( groups1681761925125756287l_real @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6218_prod__mono,axiom,
    ! [A4: set_nat,F2: nat > real,G2: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ ( groups129246275422532515t_real @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6219_prod__mono,axiom,
    ! [A4: set_int,F2: int > real,G2: int > real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( groups2316167850115554303t_real @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6220_prod__mono,axiom,
    ! [A4: set_complex,F2: complex > rat,G2: complex > rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ ( groups225925009352817453ex_rat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6221_prod__mono,axiom,
    ! [A4: set_real,F2: real > rat,G2: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) @ ( groups4061424788464935467al_rat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6222_prod__mono,axiom,
    ! [A4: set_nat,F2: nat > rat,G2: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F2 @ A4 ) @ ( groups73079841787564623at_rat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6223_prod__mono,axiom,
    ! [A4: set_int,F2: int > rat,G2: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F2 @ A4 ) @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6224_prod__mono,axiom,
    ! [A4: set_complex,F2: complex > nat,G2: complex > nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) )
            & ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups861055069439313189ex_nat @ F2 @ A4 ) @ ( groups861055069439313189ex_nat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6225_prod__mono,axiom,
    ! [A4: set_real,F2: real > nat,G2: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) )
            & ( ord_less_eq_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
     => ( ord_less_eq_nat @ ( groups4696554848551431203al_nat @ F2 @ A4 ) @ ( groups4696554848551431203al_nat @ G2 @ A4 ) ) ) ).

% prod_mono
thf(fact_6226_prod__nonneg,axiom,
    ! [A4: set_nat,F2: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F2 @ A4 ) ) ) ).

% prod_nonneg
thf(fact_6227_prod__nonneg,axiom,
    ! [A4: set_int,F2: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F2 @ A4 ) ) ) ).

% prod_nonneg
thf(fact_6228_prod__nonneg,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F2 @ A4 ) ) ) ).

% prod_nonneg
thf(fact_6229_prod__pos,axiom,
    ! [A4: set_nat,F2: nat > int] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F2 @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups705719431365010083at_int @ F2 @ A4 ) ) ) ).

% prod_pos
thf(fact_6230_prod__pos,axiom,
    ! [A4: set_int,F2: int > int] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_int @ zero_zero_int @ ( F2 @ X3 ) ) )
     => ( ord_less_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F2 @ A4 ) ) ) ).

% prod_pos
thf(fact_6231_prod__pos,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_nat @ zero_zero_nat @ ( F2 @ X3 ) ) )
     => ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F2 @ A4 ) ) ) ).

% prod_pos
thf(fact_6232_prod__ge__1,axiom,
    ! [A4: set_complex,F2: complex > code_integer] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups8682486955453173170nteger @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6233_prod__ge__1,axiom,
    ! [A4: set_real,F2: real > code_integer] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups6225526099057966256nteger @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6234_prod__ge__1,axiom,
    ! [A4: set_nat,F2: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6235_prod__ge__1,axiom,
    ! [A4: set_int,F2: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ X3 ) ) )
     => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6236_prod__ge__1,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups766887009212190081x_real @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6237_prod__ge__1,axiom,
    ! [A4: set_real,F2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6238_prod__ge__1,axiom,
    ! [A4: set_nat,F2: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups129246275422532515t_real @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6239_prod__ge__1,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ord_less_eq_real @ one_one_real @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_real @ one_one_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6240_prod__ge__1,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6241_prod__ge__1,axiom,
    ! [A4: set_real,F2: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ X3 ) ) )
     => ( ord_less_eq_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) ) ) ).

% prod_ge_1
thf(fact_6242_prod__zero,axiom,
    ! [A4: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_complex ) )
       => ( ( groups6464643781859351333omplex @ F2 @ A4 )
          = zero_zero_complex ) ) ) ).

% prod_zero
thf(fact_6243_prod__zero,axiom,
    ! [A4: set_int,F2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_complex ) )
       => ( ( groups7440179247065528705omplex @ F2 @ A4 )
          = zero_zero_complex ) ) ) ).

% prod_zero
thf(fact_6244_prod__zero,axiom,
    ! [A4: set_complex,F2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ? [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_complex ) )
       => ( ( groups3708469109370488835omplex @ F2 @ A4 )
          = zero_zero_complex ) ) ) ).

% prod_zero
thf(fact_6245_prod__zero,axiom,
    ! [A4: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_real ) )
       => ( ( groups129246275422532515t_real @ F2 @ A4 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6246_prod__zero,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_real ) )
       => ( ( groups2316167850115554303t_real @ F2 @ A4 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6247_prod__zero,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ? [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_real ) )
       => ( ( groups766887009212190081x_real @ F2 @ A4 )
          = zero_zero_real ) ) ) ).

% prod_zero
thf(fact_6248_prod__zero,axiom,
    ! [A4: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ? [X5: nat] :
            ( ( member_nat @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_rat ) )
       => ( ( groups73079841787564623at_rat @ F2 @ A4 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_6249_prod__zero,axiom,
    ! [A4: set_int,F2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_rat ) )
       => ( ( groups1072433553688619179nt_rat @ F2 @ A4 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_6250_prod__zero,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ? [X5: complex] :
            ( ( member_complex @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_rat ) )
       => ( ( groups225925009352817453ex_rat @ F2 @ A4 )
          = zero_zero_rat ) ) ) ).

% prod_zero
thf(fact_6251_prod__zero,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ? [X5: int] :
            ( ( member_int @ X5 @ A4 )
            & ( ( F2 @ X5 )
              = zero_zero_nat ) )
       => ( ( groups1707563613775114915nt_nat @ F2 @ A4 )
          = zero_zero_nat ) ) ) ).

% prod_zero
thf(fact_6252_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > code_integer,A: nat,B: nat] :
      ( ( groups3455450783089532116nteger @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1084959871951514735nteger
        @ ^ [A5: nat] : ( times_3573771949741848930nteger @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_Code_integer ) ) ).

% prod_atLeastAtMost_code
thf(fact_6253_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > complex,A: nat,B: nat] :
      ( ( groups6464643781859351333omplex @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( times_times_complex @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_complex ) ) ).

% prod_atLeastAtMost_code
thf(fact_6254_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > real,A: nat,B: nat] :
      ( ( groups129246275422532515t_real @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( times_times_real @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_real ) ) ).

% prod_atLeastAtMost_code
thf(fact_6255_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > rat,A: nat,B: nat] :
      ( ( groups73079841787564623at_rat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( times_times_rat @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_rat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6256_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > int,A: nat,B: nat] :
      ( ( groups705719431365010083at_int @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( times_times_int @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_int ) ) ).

% prod_atLeastAtMost_code
thf(fact_6257_prod__atLeastAtMost__code,axiom,
    ! [F2: nat > nat,A: nat,B: nat] :
      ( ( groups708209901874060359at_nat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( times_times_nat @ ( F2 @ A5 ) )
        @ A
        @ B
        @ one_one_nat ) ) ).

% prod_atLeastAtMost_code
thf(fact_6258_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_6259_lessThan__strict__subset__iff,axiom,
    ! [M2: real,N: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N ) )
      = ( ord_less_real @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6260_lessThan__strict__subset__iff,axiom,
    ! [M2: rat,N: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M2 ) @ ( set_ord_lessThan_rat @ N ) )
      = ( ord_less_rat @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6261_lessThan__strict__subset__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M2 ) @ ( set_ord_lessThan_num @ N ) )
      = ( ord_less_num @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6262_lessThan__strict__subset__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M2 ) @ ( set_ord_lessThan_int @ N ) )
      = ( ord_less_int @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6263_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_6264_lessThan__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( insert_nat @ K2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% lessThan_Suc
thf(fact_6265_lessThan__empty__iff,axiom,
    ! [N: nat] :
      ( ( ( set_ord_lessThan_nat @ N )
        = bot_bot_set_nat )
      = ( N = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_6266_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X4: real] : ( plus_plus_real @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_6267_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X4: rat] : ( plus_plus_rat @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_6268_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X4: int] : ( plus_plus_int @ X4 @ X4 ) ) ) ).

% dbl_def
thf(fact_6269_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6270_prod_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_Suc_ivl
thf(fact_6271_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > int,M2: nat,K2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G2 @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6272_prod_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G2: nat > nat,M2: nat,K2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G2 @ ( plus_plus_nat @ I @ K2 ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.shift_bounds_cl_nat_ivl
thf(fact_6273_prod__le__1,axiom,
    ! [A4: set_complex,F2: complex > code_integer] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups8682486955453173170nteger @ F2 @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6274_prod__le__1,axiom,
    ! [A4: set_real,F2: real > code_integer] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups6225526099057966256nteger @ F2 @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6275_prod__le__1,axiom,
    ! [A4: set_nat,F2: nat > code_integer] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F2 @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6276_prod__le__1,axiom,
    ! [A4: set_int,F2: int > code_integer] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ X3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ X3 ) @ one_one_Code_integer ) ) )
     => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F2 @ A4 ) @ one_one_Code_integer ) ) ).

% prod_le_1
thf(fact_6277_prod__le__1,axiom,
    ! [A4: set_complex,F2: complex > real] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) )
            & ( ord_less_eq_real @ ( F2 @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6278_prod__le__1,axiom,
    ! [A4: set_real,F2: real > real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) )
            & ( ord_less_eq_real @ ( F2 @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6279_prod__le__1,axiom,
    ! [A4: set_nat,F2: nat > real] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) )
            & ( ord_less_eq_real @ ( F2 @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6280_prod__le__1,axiom,
    ! [A4: set_int,F2: int > real] :
      ( ! [X3: int] :
          ( ( member_int @ X3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ X3 ) )
            & ( ord_less_eq_real @ ( F2 @ X3 ) @ one_one_real ) ) )
     => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ one_one_real ) ) ).

% prod_le_1
thf(fact_6281_prod__le__1,axiom,
    ! [A4: set_complex,F2: complex > rat] :
      ( ! [X3: complex] :
          ( ( member_complex @ X3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) )
            & ( ord_less_eq_rat @ ( F2 @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6282_prod__le__1,axiom,
    ! [A4: set_real,F2: real > rat] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ X3 ) )
            & ( ord_less_eq_rat @ ( F2 @ X3 ) @ one_one_rat ) ) )
     => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) @ one_one_rat ) ) ).

% prod_le_1
thf(fact_6283_prod_Orelated,axiom,
    ! [R2: code_integer > code_integer > $o,S3: set_nat,H: nat > code_integer,G2: nat > code_integer] :
      ( ( R2 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_3573771949741848930nteger @ X15 @ Y15 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups3455450783089532116nteger @ H @ S3 ) @ ( groups3455450783089532116nteger @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6284_prod_Orelated,axiom,
    ! [R2: code_integer > code_integer > $o,S3: set_int,H: int > code_integer,G2: int > code_integer] :
      ( ( R2 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_3573771949741848930nteger @ X15 @ Y15 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups3827104343326376752nteger @ H @ S3 ) @ ( groups3827104343326376752nteger @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6285_prod_Orelated,axiom,
    ! [R2: code_integer > code_integer > $o,S3: set_complex,H: complex > code_integer,G2: complex > code_integer] :
      ( ( R2 @ one_one_Code_integer @ one_one_Code_integer )
     => ( ! [X15: code_integer,Y15: code_integer,X23: code_integer,Y23: code_integer] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_3573771949741848930nteger @ X15 @ Y15 ) @ ( times_3573771949741848930nteger @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups8682486955453173170nteger @ H @ S3 ) @ ( groups8682486955453173170nteger @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6286_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_nat,H: nat > complex,G2: nat > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups6464643781859351333omplex @ H @ S3 ) @ ( groups6464643781859351333omplex @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6287_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_int,H: int > complex,G2: int > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups7440179247065528705omplex @ H @ S3 ) @ ( groups7440179247065528705omplex @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6288_prod_Orelated,axiom,
    ! [R2: complex > complex > $o,S3: set_complex,H: complex > complex,G2: complex > complex] :
      ( ( R2 @ one_one_complex @ one_one_complex )
     => ( ! [X15: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_complex @ X15 @ Y15 ) @ ( times_times_complex @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups3708469109370488835omplex @ H @ S3 ) @ ( groups3708469109370488835omplex @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6289_prod_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_nat,H: nat > real,G2: nat > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups129246275422532515t_real @ H @ S3 ) @ ( groups129246275422532515t_real @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6290_prod_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_int,H: int > real,G2: int > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X3: int] :
                ( ( member_int @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups2316167850115554303t_real @ H @ S3 ) @ ( groups2316167850115554303t_real @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6291_prod_Orelated,axiom,
    ! [R2: real > real > $o,S3: set_complex,H: complex > real,G2: complex > real] :
      ( ( R2 @ one_one_real @ one_one_real )
     => ( ! [X15: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_real @ X15 @ Y15 ) @ ( times_times_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X3: complex] :
                ( ( member_complex @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups766887009212190081x_real @ H @ S3 ) @ ( groups766887009212190081x_real @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6292_prod_Orelated,axiom,
    ! [R2: rat > rat > $o,S3: set_nat,H: nat > rat,G2: nat > rat] :
      ( ( R2 @ one_one_rat @ one_one_rat )
     => ( ! [X15: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R2 @ X15 @ X23 )
              & ( R2 @ Y15 @ Y23 ) )
           => ( R2 @ ( times_times_rat @ X15 @ Y15 ) @ ( times_times_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X3: nat] :
                ( ( member_nat @ X3 @ S3 )
               => ( R2 @ ( H @ X3 ) @ ( G2 @ X3 ) ) )
           => ( R2 @ ( groups73079841787564623at_rat @ H @ S3 ) @ ( groups73079841787564623at_rat @ G2 @ S3 ) ) ) ) ) ) ).

% prod.related
thf(fact_6293_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A4 ) )
            = ( groups1681761925125756287l_real @ G2 @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A4 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6294_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A4 ) )
            = ( groups129246275422532515t_real @ G2 @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A4 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6295_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A4 ) )
            = ( groups2316167850115554303t_real @ G2 @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A4 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6296_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X @ A4 )
         => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A4 ) )
            = ( groups766887009212190081x_real @ G2 @ A4 ) ) )
        & ( ~ ( member_complex @ X @ A4 )
         => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A4 ) )
            = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6297_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A4 ) )
            = ( groups4061424788464935467al_rat @ G2 @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A4 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6298_prod_Oinsert__if,axiom,
    ! [A4: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( member_nat @ X @ A4 )
         => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A4 ) )
            = ( groups73079841787564623at_rat @ G2 @ A4 ) ) )
        & ( ~ ( member_nat @ X @ A4 )
         => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A4 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6299_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A4 ) )
            = ( groups1072433553688619179nt_rat @ G2 @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A4 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6300_prod_Oinsert__if,axiom,
    ! [A4: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( member_complex @ X @ A4 )
         => ( ( groups225925009352817453ex_rat @ G2 @ ( insert_complex @ X @ A4 ) )
            = ( groups225925009352817453ex_rat @ G2 @ A4 ) ) )
        & ( ~ ( member_complex @ X @ A4 )
         => ( ( groups225925009352817453ex_rat @ G2 @ ( insert_complex @ X @ A4 ) )
            = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6301_prod_Oinsert__if,axiom,
    ! [A4: set_real,X: real,G2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( member_real @ X @ A4 )
         => ( ( groups4696554848551431203al_nat @ G2 @ ( insert_real @ X @ A4 ) )
            = ( groups4696554848551431203al_nat @ G2 @ A4 ) ) )
        & ( ~ ( member_real @ X @ A4 )
         => ( ( groups4696554848551431203al_nat @ G2 @ ( insert_real @ X @ A4 ) )
            = ( times_times_nat @ ( G2 @ X ) @ ( groups4696554848551431203al_nat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6302_prod_Oinsert__if,axiom,
    ! [A4: set_int,X: int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( member_int @ X @ A4 )
         => ( ( groups1707563613775114915nt_nat @ G2 @ ( insert_int @ X @ A4 ) )
            = ( groups1707563613775114915nt_nat @ G2 @ A4 ) ) )
        & ( ~ ( member_int @ X @ A4 )
         => ( ( groups1707563613775114915nt_nat @ G2 @ ( insert_int @ X @ A4 ) )
            = ( times_times_nat @ ( G2 @ X ) @ ( groups1707563613775114915nt_nat @ G2 @ A4 ) ) ) ) ) ) ).

% prod.insert_if
thf(fact_6303_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_6304_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_6305_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_6306_prod_OatLeastAtMost__rev,axiom,
    ! [G2: nat > int,N: nat,M2: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6307_prod_OatLeastAtMost__rev,axiom,
    ! [G2: nat > nat,N: nat,M2: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ N @ M2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ I ) )
        @ ( set_or1269000886237332187st_nat @ N @ M2 ) ) ) ).

% prod.atLeastAtMost_rev
thf(fact_6308_sum_Onat__diff__reindex,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6309_sum_Onat__diff__reindex,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( G2 @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
        @ ( set_ord_lessThan_nat @ N ) )
      = ( groups6591440286371151544t_real @ G2 @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nat_diff_reindex
thf(fact_6310_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P2: nat > nat,N: nat] :
      ( ! [X3: nat] : ( ord_less_eq_nat @ ( Q @ X3 ) @ ( P2 @ X3 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P2 @ ( set_ord_lessThan_nat @ N ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X4: nat] : ( minus_minus_nat @ ( P2 @ X4 ) @ ( Q @ X4 ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_diff_distrib
thf(fact_6311_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: real > code_integer] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups6225526099057966256nteger @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6312_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: nat > code_integer] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3455450783089532116nteger @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6313_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F2: int > code_integer] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups3827104343326376752nteger @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6314_less__1__prod2,axiom,
    ! [I5: set_complex,I2: complex,F2: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I2 @ I5 )
       => ( ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( F2 @ I2 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ I3 ) ) )
           => ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( groups8682486955453173170nteger @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6315_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups1681761925125756287l_real @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6316_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: nat > real] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F2 @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups129246275422532515t_real @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6317_less__1__prod2,axiom,
    ! [I5: set_int,I2: int,F2: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I2 @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F2 @ I2 ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups2316167850115554303t_real @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6318_less__1__prod2,axiom,
    ! [I5: set_complex,I2: complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I2 @ I5 )
       => ( ( ord_less_real @ one_one_real @ ( F2 @ I2 ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ one_one_real @ ( F2 @ I3 ) ) )
           => ( ord_less_real @ one_one_real @ ( groups766887009212190081x_real @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6319_less__1__prod2,axiom,
    ! [I5: set_real,I2: real,F2: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F2 @ I2 ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups4061424788464935467al_rat @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6320_less__1__prod2,axiom,
    ! [I5: set_nat,I2: nat,F2: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I2 @ I5 )
       => ( ( ord_less_rat @ one_one_rat @ ( F2 @ I2 ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ I3 ) ) )
           => ( ord_less_rat @ one_one_rat @ ( groups73079841787564623at_rat @ F2 @ I5 ) ) ) ) ) ) ).

% less_1_prod2
thf(fact_6321_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: int > real] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups2316167850115554303t_real @ G2 @ A4 )
          = ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6322_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: complex > real] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups766887009212190081x_real @ G2 @ A4 )
          = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6323_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: int > rat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G2 @ A4 )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6324_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups225925009352817453ex_rat @ G2 @ A4 )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6325_prod_Osubset__diff,axiom,
    ! [B5: set_int,A4: set_int,G2: int > nat] :
      ( ( ord_less_eq_set_int @ B5 @ A4 )
     => ( ( finite_finite_int @ A4 )
       => ( ( groups1707563613775114915nt_nat @ G2 @ A4 )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6326_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups861055069439313189ex_nat @ G2 @ A4 )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6327_prod_Osubset__diff,axiom,
    ! [B5: set_complex,A4: set_complex,G2: complex > int] :
      ( ( ord_le211207098394363844omplex @ B5 @ A4 )
     => ( ( finite3207457112153483333omplex @ A4 )
       => ( ( groups858564598930262913ex_int @ G2 @ A4 )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6328_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: nat > real] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups129246275422532515t_real @ G2 @ A4 )
          = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6329_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: nat > rat] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups73079841787564623at_rat @ G2 @ A4 )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6330_prod_Osubset__diff,axiom,
    ! [B5: set_nat,A4: set_nat,G2: nat > int] :
      ( ( ord_less_eq_set_nat @ B5 @ A4 )
     => ( ( finite_finite_nat @ A4 )
       => ( ( groups705719431365010083at_int @ G2 @ A4 )
          = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups705719431365010083at_int @ G2 @ B5 ) ) ) ) ) ).

% prod.subset_diff
thf(fact_6331_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ A4 ) @ ( groups2316167850115554303t_real @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6332_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( times_times_real @ ( groups766887009212190081x_real @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) )
          = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ A4 ) @ ( groups766887009212190081x_real @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6333_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ A4 ) @ ( groups129246275422532515t_real @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6334_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) @ ( groups1072433553688619179nt_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6335_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ A4 ) @ ( groups225925009352817453ex_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6336_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ A4 ) @ ( groups73079841787564623at_rat @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6337_prod_Ounion__inter,axiom,
    ! [A4: set_int,B5: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) )
          = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ A4 ) @ ( groups1707563613775114915nt_nat @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6338_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) )
          = ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ A4 ) @ ( groups861055069439313189ex_nat @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6339_prod_Ounion__inter,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) )
          = ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ A4 ) @ ( groups858564598930262913ex_int @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6340_prod_Ounion__inter,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) ) @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ A4 ) @ ( groups705719431365010083at_int @ G2 @ B5 ) ) ) ) ) ).

% prod.union_inter
thf(fact_6341_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: int > real,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups2316167850115554303t_real @ G2 @ A4 )
        = ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6342_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: complex > real,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G2 @ A4 )
        = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6343_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: nat > real,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real @ G2 @ A4 )
        = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6344_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: int > rat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G2 @ A4 )
        = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6345_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: complex > rat,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat @ G2 @ A4 )
        = ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6346_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: nat > rat,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G2 @ A4 )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6347_prod_OInt__Diff,axiom,
    ! [A4: set_int,G2: int > nat,B5: set_int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1707563613775114915nt_nat @ G2 @ A4 )
        = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6348_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: complex > nat,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups861055069439313189ex_nat @ G2 @ A4 )
        = ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6349_prod_OInt__Diff,axiom,
    ! [A4: set_complex,G2: complex > int,B5: set_complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups858564598930262913ex_int @ G2 @ A4 )
        = ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6350_prod_OInt__Diff,axiom,
    ! [A4: set_nat,G2: nat > int,B5: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups705719431365010083at_int @ G2 @ A4 )
        = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) ) ) ) ).

% prod.Int_Diff
thf(fact_6351_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6352_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6353_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6354_prod_OatLeast0__atMost__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atLeast0_atMost_Suc
thf(fact_6355_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( G2 @ M2 ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6356_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( G2 @ M2 ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6357_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( G2 @ M2 ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6358_prod_OatLeast__Suc__atMost,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_nat @ ( G2 @ M2 ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ N ) ) ) ) ) ).

% prod.atLeast_Suc_atMost
thf(fact_6359_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G2 @ ( suc @ N ) ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6360_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G2 @ ( suc @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6361_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G2 @ ( suc @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6362_prod_Onat__ivl__Suc_H,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G2 @ ( suc @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.nat_ivl_Suc'
thf(fact_6363_Iio__Int__singleton,axiom,
    ! [X: real,K2: real] :
      ( ( ( ord_less_real @ X @ K2 )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
          = ( insert_real @ X @ bot_bot_set_real ) ) )
      & ( ~ ( ord_less_real @ X @ K2 )
       => ( ( inf_inf_set_real @ ( set_or5984915006950818249n_real @ K2 ) @ ( insert_real @ X @ bot_bot_set_real ) )
          = bot_bot_set_real ) ) ) ).

% Iio_Int_singleton
thf(fact_6364_Iio__Int__singleton,axiom,
    ! [X: rat,K2: rat] :
      ( ( ( ord_less_rat @ X @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
          = ( insert_rat @ X @ bot_bot_set_rat ) ) )
      & ( ~ ( ord_less_rat @ X @ K2 )
       => ( ( inf_inf_set_rat @ ( set_ord_lessThan_rat @ K2 ) @ ( insert_rat @ X @ bot_bot_set_rat ) )
          = bot_bot_set_rat ) ) ) ).

% Iio_Int_singleton
thf(fact_6365_Iio__Int__singleton,axiom,
    ! [X: num,K2: num] :
      ( ( ( ord_less_num @ X @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X @ bot_bot_set_num ) )
          = ( insert_num @ X @ bot_bot_set_num ) ) )
      & ( ~ ( ord_less_num @ X @ K2 )
       => ( ( inf_inf_set_num @ ( set_ord_lessThan_num @ K2 ) @ ( insert_num @ X @ bot_bot_set_num ) )
          = bot_bot_set_num ) ) ) ).

% Iio_Int_singleton
thf(fact_6366_Iio__Int__singleton,axiom,
    ! [X: int,K2: int] :
      ( ( ( ord_less_int @ X @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
          = ( insert_int @ X @ bot_bot_set_int ) ) )
      & ( ~ ( ord_less_int @ X @ K2 )
       => ( ( inf_inf_set_int @ ( set_ord_lessThan_int @ K2 ) @ ( insert_int @ X @ bot_bot_set_int ) )
          = bot_bot_set_int ) ) ) ).

% Iio_Int_singleton
thf(fact_6367_Iio__Int__singleton,axiom,
    ! [X: nat,K2: nat] :
      ( ( ( ord_less_nat @ X @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = ( insert_nat @ X @ bot_bot_set_nat ) ) )
      & ( ~ ( ord_less_nat @ X @ K2 )
       => ( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ K2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
          = bot_bot_set_nat ) ) ) ).

% Iio_Int_singleton
thf(fact_6368_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_real @ ( G2 @ M2 )
          @ ( groups129246275422532515t_real
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6369_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_rat @ ( G2 @ M2 )
          @ ( groups73079841787564623at_rat
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6370_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_int @ ( G2 @ M2 )
          @ ( groups705719431365010083at_int
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6371_prod_OSuc__reindex__ivl,axiom,
    ! [M2: nat,N: nat,G2: nat > nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( G2 @ ( suc @ N ) ) )
        = ( times_times_nat @ ( G2 @ M2 )
          @ ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
            @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ) ) ).

% prod.Suc_reindex_ivl
thf(fact_6372_sum_OlessThan__Suc__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G2 @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6373_sum_OlessThan__Suc__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G2 @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6374_sum_OlessThan__Suc__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G2 @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6375_sum_OlessThan__Suc__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G2 @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_6376_sum__lessThan__telescope_H,axiom,
    ! [F2: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F2 @ zero_zero_nat ) @ ( F2 @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6377_sum__lessThan__telescope_H,axiom,
    ! [F2: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F2 @ zero_zero_nat ) @ ( F2 @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6378_sum__lessThan__telescope_H,axiom,
    ! [F2: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( minus_minus_real @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F2 @ zero_zero_nat ) @ ( F2 @ M2 ) ) ) ).

% sum_lessThan_telescope'
thf(fact_6379_sum__lessThan__telescope,axiom,
    ! [F2: nat > rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N4: nat] : ( minus_minus_rat @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_rat @ ( F2 @ M2 ) @ ( F2 @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6380_sum__lessThan__telescope,axiom,
    ! [F2: nat > int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N4: nat] : ( minus_minus_int @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_int @ ( F2 @ M2 ) @ ( F2 @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6381_sum__lessThan__telescope,axiom,
    ! [F2: nat > real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N4: nat] : ( minus_minus_real @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( minus_minus_real @ ( F2 @ M2 ) @ ( F2 @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_6382_sumr__diff__mult__const2,axiom,
    ! [F2: nat > rat,N: nat,R3: rat] :
      ( ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F2 @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ R3 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( minus_minus_rat @ ( F2 @ I ) @ R3 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6383_sumr__diff__mult__const2,axiom,
    ! [F2: nat > int,N: nat,R3: int] :
      ( ( minus_minus_int @ ( groups3539618377306564664at_int @ F2 @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ R3 ) )
      = ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( minus_minus_int @ ( F2 @ I ) @ R3 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6384_sumr__diff__mult__const2,axiom,
    ! [F2: nat > real,N: nat,R3: real] :
      ( ( minus_minus_real @ ( groups6591440286371151544t_real @ F2 @ ( set_ord_lessThan_nat @ N ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ R3 ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( minus_minus_real @ ( F2 @ I ) @ R3 )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sumr_diff_mult_const2
thf(fact_6385_sum_OatLeast1__atMost__eq,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6386_sum_OatLeast1__atMost__eq,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_6387_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
              & ( ord_less_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( groups766887009212190081x_real @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6388_prod__mono__strict,axiom,
    ! [A4: set_nat,F2: nat > real,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
              & ( ord_less_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ ( groups129246275422532515t_real @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6389_prod__mono__strict,axiom,
    ! [A4: set_int,F2: int > real,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
              & ( ord_less_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( groups2316167850115554303t_real @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6390_prod__mono__strict,axiom,
    ! [A4: set_real,F2: real > real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ A4 )
           => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
              & ( ord_less_real @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ ( groups1681761925125756287l_real @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6391_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
              & ( ord_less_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ ( groups225925009352817453ex_rat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6392_prod__mono__strict,axiom,
    ! [A4: set_nat,F2: nat > rat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
              & ( ord_less_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_nat )
         => ( ord_less_rat @ ( groups73079841787564623at_rat @ F2 @ A4 ) @ ( groups73079841787564623at_rat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6393_prod__mono__strict,axiom,
    ! [A4: set_int,F2: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
              & ( ord_less_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_rat @ ( groups1072433553688619179nt_rat @ F2 @ A4 ) @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6394_prod__mono__strict,axiom,
    ! [A4: set_real,F2: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ A4 )
           => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
              & ( ord_less_rat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_real )
         => ( ord_less_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) @ ( groups4061424788464935467al_rat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6395_prod__mono__strict,axiom,
    ! [A4: set_complex,F2: complex > nat,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) )
              & ( ord_less_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_complex )
         => ( ord_less_nat @ ( groups861055069439313189ex_nat @ F2 @ A4 ) @ ( groups861055069439313189ex_nat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6396_prod__mono__strict,axiom,
    ! [A4: set_int,F2: int > nat,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ A4 )
           => ( ( ord_less_eq_nat @ zero_zero_nat @ ( F2 @ I3 ) )
              & ( ord_less_nat @ ( F2 @ I3 ) @ ( G2 @ I3 ) ) ) )
       => ( ( A4 != bot_bot_set_int )
         => ( ord_less_nat @ ( groups1707563613775114915nt_nat @ F2 @ A4 ) @ ( groups1707563613775114915nt_nat @ G2 @ A4 ) ) ) ) ) ).

% prod_mono_strict
thf(fact_6397_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( groups766887009212190081x_real @ G2 @ A4 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6398_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( groups2316167850115554303t_real @ G2 @ A4 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6399_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( groups1681761925125756287l_real @ G2 @ A4 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6400_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( groups129246275422532515t_real @ G2 @ A4 )
          = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6401_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( groups225925009352817453ex_rat @ G2 @ A4 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6402_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( groups1072433553688619179nt_rat @ G2 @ A4 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6403_prod_Oremove,axiom,
    ! [A4: set_real,X: real,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( member_real @ X @ A4 )
       => ( ( groups4061424788464935467al_rat @ G2 @ A4 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6404_prod_Oremove,axiom,
    ! [A4: set_nat,X: nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( member_nat @ X @ A4 )
       => ( ( groups73079841787564623at_rat @ G2 @ A4 )
          = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6405_prod_Oremove,axiom,
    ! [A4: set_complex,X: complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( member_complex @ X @ A4 )
       => ( ( groups861055069439313189ex_nat @ G2 @ A4 )
          = ( times_times_nat @ ( G2 @ X ) @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6406_prod_Oremove,axiom,
    ! [A4: set_int,X: int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( member_int @ X @ A4 )
       => ( ( groups1707563613775114915nt_nat @ G2 @ A4 )
          = ( times_times_nat @ ( G2 @ X ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.remove
thf(fact_6407_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: complex > real,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real @ G2 @ ( insert_complex @ X @ A4 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6408_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: int > real,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups2316167850115554303t_real @ G2 @ ( insert_int @ X @ A4 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6409_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G2: real > real,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1681761925125756287l_real @ G2 @ ( insert_real @ X @ A4 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups1681761925125756287l_real @ G2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6410_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G2: nat > real,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real @ G2 @ ( insert_nat @ X @ A4 ) )
        = ( times_times_real @ ( G2 @ X ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6411_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: complex > rat,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat @ G2 @ ( insert_complex @ X @ A4 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6412_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: int > rat,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat @ G2 @ ( insert_int @ X @ A4 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6413_prod_Oinsert__remove,axiom,
    ! [A4: set_real,G2: real > rat,X: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4061424788464935467al_rat @ G2 @ ( insert_real @ X @ A4 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups4061424788464935467al_rat @ G2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ X @ bot_bot_set_real ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6414_prod_Oinsert__remove,axiom,
    ! [A4: set_nat,G2: nat > rat,X: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat @ G2 @ ( insert_nat @ X @ A4 ) )
        = ( times_times_rat @ ( G2 @ X ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6415_prod_Oinsert__remove,axiom,
    ! [A4: set_complex,G2: complex > nat,X: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups861055069439313189ex_nat @ G2 @ ( insert_complex @ X @ A4 ) )
        = ( times_times_nat @ ( G2 @ X ) @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ X @ bot_bot_set_complex ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6416_prod_Oinsert__remove,axiom,
    ! [A4: set_int,G2: int > nat,X: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1707563613775114915nt_nat @ G2 @ ( insert_int @ X @ A4 ) )
        = ( times_times_nat @ ( G2 @ X ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ).

% prod.insert_remove
thf(fact_6417_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > code_integer] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3827104343326376752nteger @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups3827104343326376752nteger @ G2 @ A4 ) @ ( groups3827104343326376752nteger @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6418_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups8682486955453173170nteger @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups8682486955453173170nteger @ G2 @ A4 ) @ ( groups8682486955453173170nteger @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6419_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_complex ) )
         => ( ( groups7440179247065528705omplex @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_complex @ ( groups7440179247065528705omplex @ G2 @ A4 ) @ ( groups7440179247065528705omplex @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6420_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_complex ) )
         => ( ( groups3708469109370488835omplex @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_times_complex @ ( groups3708469109370488835omplex @ G2 @ A4 ) @ ( groups3708469109370488835omplex @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6421_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > code_integer] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_Code_integer ) )
         => ( ( groups3455450783089532116nteger @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_3573771949741848930nteger @ ( groups3455450783089532116nteger @ G2 @ A4 ) @ ( groups3455450783089532116nteger @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6422_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_complex ) )
         => ( ( groups6464643781859351333omplex @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_complex @ ( groups6464643781859351333omplex @ G2 @ A4 ) @ ( groups6464643781859351333omplex @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6423_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_real ) )
         => ( ( groups2316167850115554303t_real @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ A4 ) @ ( groups2316167850115554303t_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6424_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_real ) )
         => ( ( groups766887009212190081x_real @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ A4 ) @ ( groups766887009212190081x_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6425_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_real ) )
         => ( ( groups129246275422532515t_real @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ A4 ) @ ( groups129246275422532515t_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6426_prod_Ounion__inter__neutral,axiom,
    ! [A4: set_int,B5: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( G2 @ X3 )
                = one_one_rat ) )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) @ ( groups1072433553688619179nt_rat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_inter_neutral
thf(fact_6427_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( ( inf_inf_set_complex @ A4 @ B5 )
            = bot_bot_set_complex )
         => ( ( groups766887009212190081x_real @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_times_real @ ( groups766887009212190081x_real @ G2 @ A4 ) @ ( groups766887009212190081x_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6428_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups129246275422532515t_real @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ A4 ) @ ( groups129246275422532515t_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6429_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups2316167850115554303t_real @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ A4 ) @ ( groups2316167850115554303t_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6430_prod_Ounion__disjoint,axiom,
    ! [A4: set_real,B5: set_real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite_finite_real @ B5 )
       => ( ( ( inf_inf_set_real @ A4 @ B5 )
            = bot_bot_set_real )
         => ( ( groups1681761925125756287l_real @ G2 @ ( sup_sup_set_real @ A4 @ B5 ) )
            = ( times_times_real @ ( groups1681761925125756287l_real @ G2 @ A4 ) @ ( groups1681761925125756287l_real @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6431_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( ( inf_inf_set_complex @ A4 @ B5 )
            = bot_bot_set_complex )
         => ( ( groups225925009352817453ex_rat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ A4 ) @ ( groups225925009352817453ex_rat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6432_prod_Ounion__disjoint,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( ( inf_inf_set_nat @ A4 @ B5 )
            = bot_bot_set_nat )
         => ( ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ A4 ) @ ( groups73079841787564623at_rat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6433_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ A4 ) @ ( groups1072433553688619179nt_rat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6434_prod_Ounion__disjoint,axiom,
    ! [A4: set_real,B5: set_real,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( finite_finite_real @ B5 )
       => ( ( ( inf_inf_set_real @ A4 @ B5 )
            = bot_bot_set_real )
         => ( ( groups4061424788464935467al_rat @ G2 @ ( sup_sup_set_real @ A4 @ B5 ) )
            = ( times_times_rat @ ( groups4061424788464935467al_rat @ G2 @ A4 ) @ ( groups4061424788464935467al_rat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6435_prod_Ounion__disjoint,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( ( inf_inf_set_complex @ A4 @ B5 )
            = bot_bot_set_complex )
         => ( ( groups861055069439313189ex_nat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ A4 ) @ ( groups861055069439313189ex_nat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6436_prod_Ounion__disjoint,axiom,
    ! [A4: set_int,B5: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( ( inf_inf_set_int @ A4 @ B5 )
            = bot_bot_set_int )
         => ( ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ A4 ) @ ( groups1707563613775114915nt_nat @ G2 @ B5 ) ) ) ) ) ) ).

% prod.union_disjoint
thf(fact_6437_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_6438_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_6439_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_6440_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_6441_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups2316167850115554303t_real @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_real @ ( times_times_real @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups2316167850115554303t_real @ G2 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups2316167850115554303t_real @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6442_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups766887009212190081x_real @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
          = ( times_times_real @ ( times_times_real @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups766887009212190081x_real @ G2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) @ ( groups766887009212190081x_real @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6443_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups129246275422532515t_real @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_real @ ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups129246275422532515t_real @ G2 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups129246275422532515t_real @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6444_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups1072433553688619179nt_rat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6445_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups225925009352817453ex_rat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups225925009352817453ex_rat @ G2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) @ ( groups225925009352817453ex_rat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6446_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups73079841787564623at_rat @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups73079841787564623at_rat @ G2 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6447_prod_Ounion__diff2,axiom,
    ! [A4: set_int,B5: set_int,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ( groups1707563613775114915nt_nat @ G2 @ ( sup_sup_set_int @ A4 @ B5 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ A4 @ B5 ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( minus_minus_set_int @ B5 @ A4 ) ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6448_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups861055069439313189ex_nat @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
          = ( times_times_nat @ ( times_times_nat @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups861055069439313189ex_nat @ G2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) @ ( groups861055069439313189ex_nat @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6449_prod_Ounion__diff2,axiom,
    ! [A4: set_complex,B5: set_complex,G2: complex > int] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ( groups858564598930262913ex_int @ G2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
          = ( times_times_int @ ( times_times_int @ ( groups858564598930262913ex_int @ G2 @ ( minus_811609699411566653omplex @ A4 @ B5 ) ) @ ( groups858564598930262913ex_int @ G2 @ ( minus_811609699411566653omplex @ B5 @ A4 ) ) ) @ ( groups858564598930262913ex_int @ G2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6450_prod_Ounion__diff2,axiom,
    ! [A4: set_nat,B5: set_nat,G2: nat > int] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ( groups705719431365010083at_int @ G2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
          = ( times_times_int @ ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ A4 @ B5 ) ) @ ( groups705719431365010083at_int @ G2 @ ( minus_minus_set_nat @ B5 @ A4 ) ) ) @ ( groups705719431365010083at_int @ G2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ).

% prod.union_diff2
thf(fact_6451_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > real,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6452_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > rat,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6453_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > int,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6454_prod_Oub__add__nat,axiom,
    ! [M2: nat,N: nat,G2: nat > nat,P: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( plus_plus_nat @ N @ one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ ( plus_plus_nat @ N @ P ) ) )
        = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) @ ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N @ one_one_nat ) @ ( plus_plus_nat @ N @ P ) ) ) ) ) ) ).

% prod.ub_add_nat
thf(fact_6455_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F4: nat > nat > nat,A5: nat,B4: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B4 @ A5 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F4 @ ( plus_plus_nat @ A5 @ one_one_nat ) @ B4 @ ( F4 @ A5 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6456_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X: nat > nat > nat,Xa2: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X @ Xa2 @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa2 )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa2 )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X @ ( plus_plus_nat @ Xa2 @ one_one_nat ) @ Xb @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6457_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C2: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups766887009212190081x_real @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups766887009212190081x_real @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6458_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > real,C2: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups2316167850115554303t_real @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups2316167850115554303t_real @ C2 @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6459_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C2: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups1681761925125756287l_real @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1681761925125756287l_real @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6460_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: nat > real,C2: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( groups129246275422532515t_real @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups129246275422532515t_real @ C2 @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6461_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C2: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups225925009352817453ex_rat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups225925009352817453ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups225925009352817453ex_rat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6462_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > rat,C2: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1072433553688619179nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups1072433553688619179nt_rat @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1072433553688619179nt_rat @ C2 @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6463_prod_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C2: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups4061424788464935467al_rat @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups4061424788464935467al_rat @ C2 @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6464_prod_Odelta__remove,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat,C2: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( groups73079841787564623at_rat @ C2 @ ( 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 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups73079841787564623at_rat @ C2 @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6465_prod_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C2: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups861055069439313189ex_nat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups861055069439313189ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups861055069439313189ex_nat @ C2 @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6466_prod_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > nat,C2: int > nat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( times_times_nat @ ( B @ A ) @ ( groups1707563613775114915nt_nat @ C2 @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups1707563613775114915nt_nat
              @ ^ [K3: int] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C2 @ K3 ) )
              @ S3 )
            = ( groups1707563613775114915nt_nat @ C2 @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% prod.delta_remove
thf(fact_6467_power__diff__1__eq,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ X @ N ) @ one_one_Code_integer )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ X @ one_one_Code_integer ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6468_power__diff__1__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6469_power__diff__1__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6470_power__diff__1__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6471_power__diff__1__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_1_eq
thf(fact_6472_one__diff__power__eq,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6473_one__diff__power__eq,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6474_one__diff__power__eq,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6475_one__diff__power__eq,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6476_one__diff__power__eq,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq
thf(fact_6477_geometric__sum,axiom,
    ! [X: complex,N: nat] :
      ( ( X != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ one_one_complex ) @ ( minus_minus_complex @ X @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_6478_geometric__sum,axiom,
    ! [X: rat,N: nat] :
      ( ( X != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ one_one_rat ) @ ( minus_minus_rat @ X @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_6479_geometric__sum,axiom,
    ! [X: real,N: nat] :
      ( ( X != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X @ N ) @ one_one_real ) @ ( minus_minus_real @ X @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_6480_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > code_integer] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups6225526099057966256nteger @ F2 @ A4 ) @ ( groups6225526099057966256nteger @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6481_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > code_integer] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F2 @ A4 ) @ ( groups3827104343326376752nteger @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6482_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( F2 @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ A3 ) ) )
           => ( ord_le3102999989581377725nteger @ ( groups8682486955453173170nteger @ F2 @ A4 ) @ ( groups8682486955453173170nteger @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6483_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > real] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F2 @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ ( groups1681761925125756287l_real @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6484_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > real] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F2 @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( groups2316167850115554303t_real @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6485_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_real @ one_one_real @ ( F2 @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( groups766887009212190081x_real @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6486_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > rat] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) @ ( groups4061424788464935467al_rat @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6487_prod__mono2,axiom,
    ! [B5: set_int,A4: set_int,F2: int > rat] :
      ( ( finite_finite_int @ B5 )
     => ( ( ord_less_eq_set_int @ A4 @ B5 )
       => ( ! [B3: int] :
              ( ( member_int @ B3 @ ( minus_minus_set_int @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B3 ) ) )
         => ( ! [A3: int] :
                ( ( member_int @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups1072433553688619179nt_rat @ F2 @ A4 ) @ ( groups1072433553688619179nt_rat @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6488_prod__mono2,axiom,
    ! [B5: set_complex,A4: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ B5 )
     => ( ( ord_le211207098394363844omplex @ A4 @ B5 )
       => ( ! [B3: complex] :
              ( ( member_complex @ B3 @ ( minus_811609699411566653omplex @ B5 @ A4 ) )
             => ( ord_less_eq_rat @ one_one_rat @ ( F2 @ B3 ) ) )
         => ( ! [A3: complex] :
                ( ( member_complex @ A3 @ A4 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ ( groups225925009352817453ex_rat @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6489_prod__mono2,axiom,
    ! [B5: set_real,A4: set_real,F2: real > int] :
      ( ( finite_finite_real @ B5 )
     => ( ( ord_less_eq_set_real @ A4 @ B5 )
       => ( ! [B3: real] :
              ( ( member_real @ B3 @ ( minus_minus_set_real @ B5 @ A4 ) )
             => ( ord_less_eq_int @ one_one_int @ ( F2 @ B3 ) ) )
         => ( ! [A3: real] :
                ( ( member_real @ A3 @ A4 )
               => ( ord_less_eq_int @ zero_zero_int @ ( F2 @ A3 ) ) )
           => ( ord_less_eq_int @ ( groups4694064378042380927al_int @ F2 @ A4 ) @ ( groups4694064378042380927al_int @ F2 @ B5 ) ) ) ) ) ) ).

% prod_mono2
thf(fact_6490_prod__le__power,axiom,
    ! [A4: set_real,F2: real > code_integer,N: code_integer,K2: nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ K2 )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N )
         => ( ord_le3102999989581377725nteger @ ( groups6225526099057966256nteger @ F2 @ A4 ) @ ( power_8256067586552552935nteger @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6491_prod__le__power,axiom,
    ! [A4: set_nat,F2: nat > code_integer,N: code_integer,K2: nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ K2 )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N )
         => ( ord_le3102999989581377725nteger @ ( groups3455450783089532116nteger @ F2 @ A4 ) @ ( power_8256067586552552935nteger @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6492_prod__le__power,axiom,
    ! [A4: set_complex,F2: complex > code_integer,N: code_integer,K2: nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ K2 )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N )
         => ( ord_le3102999989581377725nteger @ ( groups8682486955453173170nteger @ F2 @ A4 ) @ ( power_8256067586552552935nteger @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6493_prod__le__power,axiom,
    ! [A4: set_int,F2: int > code_integer,N: code_integer,K2: nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( F2 @ I3 ) )
            & ( ord_le3102999989581377725nteger @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ K2 )
       => ( ( ord_le3102999989581377725nteger @ one_one_Code_integer @ N )
         => ( ord_le3102999989581377725nteger @ ( groups3827104343326376752nteger @ F2 @ A4 ) @ ( power_8256067586552552935nteger @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6494_prod__le__power,axiom,
    ! [A4: set_real,F2: real > real,N: real,K2: nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ K2 )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ ( power_power_real @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6495_prod__le__power,axiom,
    ! [A4: set_nat,F2: nat > real,N: real,K2: nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ K2 )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ ( power_power_real @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6496_prod__le__power,axiom,
    ! [A4: set_complex,F2: complex > real,N: real,K2: nat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_complex @ A4 ) @ K2 )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( power_power_real @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6497_prod__le__power,axiom,
    ! [A4: set_int,F2: int > real,N: real,K2: nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ A4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
            & ( ord_less_eq_real @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_int @ A4 ) @ K2 )
       => ( ( ord_less_eq_real @ one_one_real @ N )
         => ( ord_less_eq_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( power_power_real @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6498_prod__le__power,axiom,
    ! [A4: set_real,F2: real > rat,N: rat,K2: nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_real @ A4 ) @ K2 )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups4061424788464935467al_rat @ F2 @ A4 ) @ ( power_power_rat @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6499_prod__le__power,axiom,
    ! [A4: set_nat,F2: nat > rat,N: rat,K2: nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ A4 )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ ( F2 @ I3 ) )
            & ( ord_less_eq_rat @ ( F2 @ I3 ) @ N ) ) )
     => ( ( ord_less_eq_nat @ ( finite_card_nat @ A4 ) @ K2 )
       => ( ( ord_less_eq_rat @ one_one_rat @ N )
         => ( ord_less_eq_rat @ ( groups73079841787564623at_rat @ F2 @ A4 ) @ ( power_power_rat @ N @ K2 ) ) ) ) ) ).

% prod_le_power
thf(fact_6500_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: int > complex] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_complex ) )
         => ( ( groups7440179247065528705omplex @ F2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( groups7440179247065528705omplex @ F2 @ A4 ) @ ( groups7440179247065528705omplex @ F2 @ B5 ) ) @ ( groups7440179247065528705omplex @ F2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6501_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: complex > complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_complex ) )
         => ( ( groups3708469109370488835omplex @ F2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( groups3708469109370488835omplex @ F2 @ A4 ) @ ( groups3708469109370488835omplex @ F2 @ B5 ) ) @ ( groups3708469109370488835omplex @ F2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6502_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: nat > complex] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_complex ) )
         => ( ( groups6464643781859351333omplex @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( groups6464643781859351333omplex @ F2 @ A4 ) @ ( groups6464643781859351333omplex @ F2 @ B5 ) ) @ ( groups6464643781859351333omplex @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6503_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_real ) )
         => ( ( groups2316167850115554303t_real @ F2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( divide_divide_real @ ( times_times_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( groups2316167850115554303t_real @ F2 @ B5 ) ) @ ( groups2316167850115554303t_real @ F2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6504_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_real ) )
         => ( ( groups766887009212190081x_real @ F2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( divide_divide_real @ ( times_times_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( groups766887009212190081x_real @ F2 @ B5 ) ) @ ( groups766887009212190081x_real @ F2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6505_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_real ) )
         => ( ( groups129246275422532515t_real @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( divide_divide_real @ ( times_times_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ ( groups129246275422532515t_real @ F2 @ B5 ) ) @ ( groups129246275422532515t_real @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6506_prod__Un,axiom,
    ! [A4: set_int,B5: set_int,F2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_finite_int @ B5 )
       => ( ! [X3: int] :
              ( ( member_int @ X3 @ ( inf_inf_set_int @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_rat ) )
         => ( ( groups1072433553688619179nt_rat @ F2 @ ( sup_sup_set_int @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups1072433553688619179nt_rat @ F2 @ A4 ) @ ( groups1072433553688619179nt_rat @ F2 @ B5 ) ) @ ( groups1072433553688619179nt_rat @ F2 @ ( inf_inf_set_int @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6507_prod__Un,axiom,
    ! [A4: set_complex,B5: set_complex,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite3207457112153483333omplex @ B5 )
       => ( ! [X3: complex] :
              ( ( member_complex @ X3 @ ( inf_inf_set_complex @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_rat ) )
         => ( ( groups225925009352817453ex_rat @ F2 @ ( sup_sup_set_complex @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ ( groups225925009352817453ex_rat @ F2 @ B5 ) ) @ ( groups225925009352817453ex_rat @ F2 @ ( inf_inf_set_complex @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6508_prod__Un,axiom,
    ! [A4: set_nat,B5: set_nat,F2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_finite_nat @ B5 )
       => ( ! [X3: nat] :
              ( ( member_nat @ X3 @ ( inf_inf_set_nat @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_rat ) )
         => ( ( groups73079841787564623at_rat @ F2 @ ( sup_sup_set_nat @ A4 @ B5 ) )
            = ( divide_divide_rat @ ( times_times_rat @ ( groups73079841787564623at_rat @ F2 @ A4 ) @ ( groups73079841787564623at_rat @ F2 @ B5 ) ) @ ( groups73079841787564623at_rat @ F2 @ ( inf_inf_set_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6509_prod__Un,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,F2: product_prod_nat_nat > complex] :
      ( ( finite6177210948735845034at_nat @ A4 )
     => ( ( finite6177210948735845034at_nat @ B5 )
       => ( ! [X3: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X3 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) )
             => ( ( F2 @ X3 )
               != zero_zero_complex ) )
         => ( ( groups8110221916422527690omplex @ F2 @ ( sup_su6327502436637775413at_nat @ A4 @ B5 ) )
            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( groups8110221916422527690omplex @ F2 @ A4 ) @ ( groups8110221916422527690omplex @ F2 @ B5 ) ) @ ( groups8110221916422527690omplex @ F2 @ ( inf_in2572325071724192079at_nat @ A4 @ B5 ) ) ) ) ) ) ) ).

% prod_Un
thf(fact_6510_prod__diff1,axiom,
    ! [A4: set_complex,F2: complex > complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_complex )
       => ( ( ( member_complex @ A @ A4 )
           => ( ( groups3708469109370488835omplex @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide1717551699836669952omplex @ ( groups3708469109370488835omplex @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A4 )
           => ( ( groups3708469109370488835omplex @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups3708469109370488835omplex @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6511_prod__diff1,axiom,
    ! [A4: set_int,F2: int > complex,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_complex )
       => ( ( ( member_int @ A @ A4 )
           => ( ( groups7440179247065528705omplex @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide1717551699836669952omplex @ ( groups7440179247065528705omplex @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_int @ A @ A4 )
           => ( ( groups7440179247065528705omplex @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups7440179247065528705omplex @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6512_prod__diff1,axiom,
    ! [A4: set_real,F2: real > complex,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_complex )
       => ( ( ( member_real @ A @ A4 )
           => ( ( groups713298508707869441omplex @ F2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide1717551699836669952omplex @ ( groups713298508707869441omplex @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_real @ A @ A4 )
           => ( ( groups713298508707869441omplex @ F2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups713298508707869441omplex @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6513_prod__diff1,axiom,
    ! [A4: set_nat,F2: nat > complex,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_complex )
       => ( ( ( member_nat @ A @ A4 )
           => ( ( groups6464643781859351333omplex @ F2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide1717551699836669952omplex @ ( groups6464643781859351333omplex @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A4 )
           => ( ( groups6464643781859351333omplex @ F2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups6464643781859351333omplex @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6514_prod__diff1,axiom,
    ! [A4: set_complex,F2: complex > real,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_real )
       => ( ( ( member_complex @ A @ A4 )
           => ( ( groups766887009212190081x_real @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_real @ ( groups766887009212190081x_real @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A4 )
           => ( ( groups766887009212190081x_real @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups766887009212190081x_real @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6515_prod__diff1,axiom,
    ! [A4: set_int,F2: int > real,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_real )
       => ( ( ( member_int @ A @ A4 )
           => ( ( groups2316167850115554303t_real @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_real @ ( groups2316167850115554303t_real @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_int @ A @ A4 )
           => ( ( groups2316167850115554303t_real @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups2316167850115554303t_real @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6516_prod__diff1,axiom,
    ! [A4: set_real,F2: real > real,A: real] :
      ( ( finite_finite_real @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_real )
       => ( ( ( member_real @ A @ A4 )
           => ( ( groups1681761925125756287l_real @ F2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( divide_divide_real @ ( groups1681761925125756287l_real @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_real @ A @ A4 )
           => ( ( groups1681761925125756287l_real @ F2 @ ( minus_minus_set_real @ A4 @ ( insert_real @ A @ bot_bot_set_real ) ) )
              = ( groups1681761925125756287l_real @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6517_prod__diff1,axiom,
    ! [A4: set_nat,F2: nat > real,A: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_real )
       => ( ( ( member_nat @ A @ A4 )
           => ( ( groups129246275422532515t_real @ F2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( divide_divide_real @ ( groups129246275422532515t_real @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_nat @ A @ A4 )
           => ( ( groups129246275422532515t_real @ F2 @ ( minus_minus_set_nat @ A4 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
              = ( groups129246275422532515t_real @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6518_prod__diff1,axiom,
    ! [A4: set_complex,F2: complex > rat,A: complex] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_rat )
       => ( ( ( member_complex @ A @ A4 )
           => ( ( groups225925009352817453ex_rat @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( divide_divide_rat @ ( groups225925009352817453ex_rat @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_complex @ A @ A4 )
           => ( ( groups225925009352817453ex_rat @ F2 @ ( minus_811609699411566653omplex @ A4 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
              = ( groups225925009352817453ex_rat @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6519_prod__diff1,axiom,
    ! [A4: set_int,F2: int > rat,A: int] :
      ( ( finite_finite_int @ A4 )
     => ( ( ( F2 @ A )
         != zero_zero_rat )
       => ( ( ( member_int @ A @ A4 )
           => ( ( groups1072433553688619179nt_rat @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( divide_divide_rat @ ( groups1072433553688619179nt_rat @ F2 @ A4 ) @ ( F2 @ A ) ) ) )
          & ( ~ ( member_int @ A @ A4 )
           => ( ( groups1072433553688619179nt_rat @ F2 @ ( minus_minus_set_int @ A4 @ ( insert_int @ A @ bot_bot_set_int ) ) )
              = ( groups1072433553688619179nt_rat @ F2 @ A4 ) ) ) ) ) ) ).

% prod_diff1
thf(fact_6520_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_6521_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_6522_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_6523_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_6524_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: real > complex,C2: complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C2 @ ( minus_minus_nat @ ( finite_card_real @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_complex @ C2 @ ( finite_card_real @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6525_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex,C2: complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C2 @ ( minus_minus_nat @ ( finite_card_nat @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_complex @ C2 @ ( finite_card_nat @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6526_prod__gen__delta,axiom,
    ! [S3: set_int,A: int,B: int > complex,C2: complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C2 @ ( minus_minus_nat @ ( finite_card_int @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_complex @ C2 @ ( finite_card_int @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6527_prod__gen__delta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex,C2: complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_complex @ ( B @ A ) @ ( power_power_complex @ C2 @ ( minus_minus_nat @ ( finite_card_complex @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_complex @ C2 @ ( finite_card_complex @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6528_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: real > real,C2: real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C2 @ ( minus_minus_nat @ ( finite_card_real @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_real @ C2 @ ( finite_card_real @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6529_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real,C2: real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C2 @ ( minus_minus_nat @ ( finite_card_nat @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_real @ C2 @ ( finite_card_nat @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6530_prod__gen__delta,axiom,
    ! [S3: set_int,A: int,B: int > real,C2: real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C2 @ ( minus_minus_nat @ ( finite_card_int @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_real @ C2 @ ( finite_card_int @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6531_prod__gen__delta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C2: real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_real @ ( B @ A ) @ ( power_power_real @ C2 @ ( minus_minus_nat @ ( finite_card_complex @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_real @ C2 @ ( finite_card_complex @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6532_prod__gen__delta,axiom,
    ! [S3: set_real,A: real,B: real > rat,C2: rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C2 @ ( minus_minus_nat @ ( finite_card_real @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_rat @ C2 @ ( finite_card_real @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6533_prod__gen__delta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat,C2: rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( times_times_rat @ ( B @ A ) @ ( power_power_rat @ C2 @ ( minus_minus_nat @ ( finite_card_nat @ S3 ) @ one_one_nat ) ) ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ C2 )
              @ S3 )
            = ( power_power_rat @ C2 @ ( finite_card_nat @ S3 ) ) ) ) ) ) ).

% prod_gen_delta
thf(fact_6534_pochhammer__Suc__prod,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6535_pochhammer__Suc__prod,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6536_pochhammer__Suc__prod,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6537_pochhammer__Suc__prod,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ I ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod
thf(fact_6538_sum__gp__strict,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri8010041392384452111omplex @ N ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6539_sum__gp__strict,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri681578069525770553at_rat @ N ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6540_sum__gp__strict,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( semiri5074537144036343181t_real @ N ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_lessThan_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp_strict
thf(fact_6541_lemma__termdiff1,axiom,
    ! [Z3: complex,H: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z3 @ P5 ) ) @ ( power_power_complex @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z3 @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_complex @ Z3 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6542_lemma__termdiff1,axiom,
    ! [Z3: rat,H: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z3 @ P5 ) ) @ ( power_power_rat @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z3 @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_rat @ Z3 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6543_lemma__termdiff1,axiom,
    ! [Z3: int,H: int,M2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z3 @ P5 ) ) @ ( power_power_int @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z3 @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_int @ Z3 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6544_lemma__termdiff1,axiom,
    ! [Z3: real,H: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z3 @ P5 ) ) @ ( power_power_real @ Z3 @ M2 ) )
        @ ( set_ord_lessThan_nat @ M2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z3 @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z3 @ H ) @ ( minus_minus_nat @ M2 @ P5 ) ) @ ( power_power_real @ Z3 @ ( minus_minus_nat @ M2 @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M2 ) ) ) ).

% lemma_termdiff1
thf(fact_6545_diff__power__eq__sum,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ ( suc @ N ) ) @ ( power_power_complex @ Y @ ( suc @ N ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6546_diff__power__eq__sum,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ ( suc @ N ) ) @ ( power_power_rat @ Y @ ( suc @ N ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6547_diff__power__eq__sum,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ ( suc @ N ) ) @ ( power_power_int @ Y @ ( suc @ N ) ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6548_diff__power__eq__sum,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ ( suc @ N ) ) @ ( power_power_real @ Y @ ( suc @ N ) ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_6549_power__diff__sumr2,axiom,
    ! [X: complex,N: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X @ N ) @ ( power_power_complex @ Y @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N @ ( suc @ I ) ) ) @ ( power_power_complex @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6550_power__diff__sumr2,axiom,
    ! [X: rat,N: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X @ N ) @ ( power_power_rat @ Y @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N @ ( suc @ I ) ) ) @ ( power_power_rat @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6551_power__diff__sumr2,axiom,
    ! [X: int,N: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
      = ( times_times_int @ ( minus_minus_int @ X @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N @ ( suc @ I ) ) ) @ ( power_power_int @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6552_power__diff__sumr2,axiom,
    ! [X: real,N: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X @ N ) @ ( power_power_real @ Y @ N ) )
      = ( times_times_real @ ( minus_minus_real @ X @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N @ ( suc @ I ) ) ) @ ( power_power_real @ X @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% power_diff_sumr2
thf(fact_6553_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: nat > rat,K5: rat,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_rat @ ( F2 @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F2 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6554_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: nat > int,K5: int,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_int @ ( F2 @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F2 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6555_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: nat > nat,K5: nat,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_nat @ ( F2 @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F2 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6556_real__sum__nat__ivl__bounded2,axiom,
    ! [N: nat,F2: nat > real,K5: real,K2: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N )
         => ( ord_less_eq_real @ ( F2 @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F2 @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ K2 ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_6557_prod_Oin__pairs,axiom,
    ! [G2: nat > real,M2: nat,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I: nat] : ( times_times_real @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6558_prod_Oin__pairs,axiom,
    ! [G2: nat > rat,M2: nat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( times_times_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6559_prod_Oin__pairs,axiom,
    ! [G2: nat > int,M2: nat,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( times_times_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6560_prod_Oin__pairs,axiom,
    ! [G2: nat > nat,M2: nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( times_times_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M2 @ N ) ) ) ).

% prod.in_pairs
thf(fact_6561_sum__atLeastAtMost__code,axiom,
    ! [F2: nat > complex,A: nat,B: nat] :
      ( ( groups2073611262835488442omplex @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A5: nat] : ( plus_plus_complex @ ( F2 @ A5 ) )
        @ A
        @ B
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_6562_sum__atLeastAtMost__code,axiom,
    ! [F2: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A5: nat] : ( plus_plus_rat @ ( F2 @ A5 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6563_sum__atLeastAtMost__code,axiom,
    ! [F2: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A5: nat] : ( plus_plus_int @ ( F2 @ A5 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_6564_sum__atLeastAtMost__code,axiom,
    ! [F2: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A5: nat] : ( plus_plus_nat @ ( F2 @ A5 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_6565_sum__atLeastAtMost__code,axiom,
    ! [F2: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F2 @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A5: nat] : ( plus_plus_real @ ( F2 @ A5 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_6566_pochhammer__Suc__prod__rev,axiom,
    ! [A: rat,N: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ ( minus_minus_nat @ N @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6567_pochhammer__Suc__prod__rev,axiom,
    ! [A: real,N: nat] :
      ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N ) )
      = ( groups129246275422532515t_real
        @ ^ [I: nat] : ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6568_pochhammer__Suc__prod__rev,axiom,
    ! [A: int,N: nat] :
      ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6569_pochhammer__Suc__prod__rev,axiom,
    ! [A: nat,N: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ N @ I ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N ) ) ) ).

% pochhammer_Suc_prod_rev
thf(fact_6570_one__diff__power__eq_H,axiom,
    ! [X: code_integer,N: nat] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ N ) )
      = ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X )
        @ ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( power_8256067586552552935nteger @ X @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6571_one__diff__power__eq_H,axiom,
    ! [X: complex,N: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ N ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X )
        @ ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( power_power_complex @ X @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6572_one__diff__power__eq_H,axiom,
    ! [X: rat,N: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ N ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( power_power_rat @ X @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6573_one__diff__power__eq_H,axiom,
    ! [X: int,N: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ N ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( power_power_int @ X @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6574_one__diff__power__eq_H,axiom,
    ! [X: real,N: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ N ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ I ) ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% one_diff_power_eq'
thf(fact_6575_sum__split__even__odd,axiom,
    ! [F2: nat > real,G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( F2 @ I ) @ ( G2 @ I ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( F2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( G2 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum_split_even_odd
thf(fact_6576_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_6577_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_6578_binomial__n__0,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6579_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_6580_binomial__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( binomial @ N @ K2 )
        = zero_zero_nat )
      = ( ord_less_nat @ N @ K2 ) ) ).

% binomial_eq_0_iff
thf(fact_6581_binomial__1,axiom,
    ! [N: nat] :
      ( ( binomial @ N @ ( suc @ zero_zero_nat ) )
      = N ) ).

% binomial_1
thf(fact_6582_binomial__0__Suc,axiom,
    ! [K2: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6583_binomial__Suc__n,axiom,
    ! [N: nat] :
      ( ( binomial @ ( suc @ N ) @ N )
      = ( suc @ N ) ) ).

% binomial_Suc_n
thf(fact_6584_prod__pos__nat__iff,axiom,
    ! [A4: set_int,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups1707563613775114915nt_nat @ F2 @ A4 ) )
        = ( ! [X4: int] :
              ( ( member_int @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6585_prod__pos__nat__iff,axiom,
    ! [A4: set_complex,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups861055069439313189ex_nat @ F2 @ A4 ) )
        = ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6586_prod__pos__nat__iff,axiom,
    ! [A4: set_nat,F2: nat > nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F2 @ A4 ) )
        = ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
             => ( ord_less_nat @ zero_zero_nat @ ( F2 @ X4 ) ) ) ) ) ) ).

% prod_pos_nat_iff
thf(fact_6587_prod__int__eq,axiom,
    ! [I2: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ) ).

% prod_int_eq
thf(fact_6588_prod__int__plus__eq,axiom,
    ! [I2: nat,J2: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J2 ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X4: int] : X4
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J2 ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_6589_binomial__eq__0,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( binomial @ N @ K2 )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6590_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_6591_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_6592_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_6593_choose__mult__lemma,axiom,
    ! [M2: nat,R3: nat,K2: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R3 ) @ K2 ) @ ( plus_plus_nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus_nat @ M2 @ K2 ) @ K2 ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M2 @ R3 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus_nat @ M2 @ R3 ) @ M2 ) ) ) ).

% choose_mult_lemma
thf(fact_6594_binomial__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( binomial @ N @ R3 ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_6595_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_6596_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_6597_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_6598_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_6599_binomial__absorb__comp,axiom,
    ! [N: nat,K2: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N @ K2 ) @ ( binomial @ N @ K2 ) )
      = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K2 ) ) ) ).

% binomial_absorb_comp
thf(fact_6600_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_6601_sum__bounds__lt__plus1,axiom,
    ! [F2: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6602_sum__bounds__lt__plus1,axiom,
    ! [F2: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F2 @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_6603_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_6604_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_6605_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_6606_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_6607_times__binomial__minus1__eq,axiom,
    ! [K2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( times_times_nat @ K2 @ ( binomial @ N @ K2 ) )
        = ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% times_binomial_minus1_eq
thf(fact_6608_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_6609_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_6610_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_6611_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_6612_choose__odd__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_odd_sum
thf(fact_6613_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_6614_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_6615_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_6616_choose__even__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N ) ) ) ).

% choose_even_sum
thf(fact_6617_card__lists__distinct__length__eq,axiom,
    ! [A4: set_list_nat,K2: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_list_nat @ A4 ) )
       => ( ( finite7325466520557071688st_nat
            @ ( collec5989764272469232197st_nat
              @ ^ [Xs: list_list_nat] :
                  ( ( ( size_s3023201423986296836st_nat @ Xs )
                    = K2 )
                  & ( distinct_list_nat @ Xs )
                  & ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_list_nat @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6618_card__lists__distinct__length__eq,axiom,
    ! [A4: set_set_nat,K2: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_set_nat @ A4 ) )
       => ( ( finite5631907774883551598et_nat
            @ ( collect_list_set_nat
              @ ^ [Xs: list_set_nat] :
                  ( ( ( size_s3254054031482475050et_nat @ Xs )
                    = K2 )
                  & ( distinct_set_nat @ Xs )
                  & ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_set_nat @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6619_card__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,K2: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_complex @ A4 ) )
       => ( ( finite5120063068150530198omplex
            @ ( collect_list_complex
              @ ^ [Xs: list_complex] :
                  ( ( ( size_s3451745648224563538omplex @ Xs )
                    = K2 )
                  & ( distinct_complex @ Xs )
                  & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_complex @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6620_card__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,K2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite7802652506058667612T_VEBT @ A4 ) )
       => ( ( finite5915292604075114978T_VEBT
            @ ( collec5608196760682091941T_VEBT
              @ ^ [Xs: list_VEBT_VEBT] :
                  ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                    = K2 )
                  & ( distinct_VEBT_VEBT @ Xs )
                  & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6621_card__lists__distinct__length__eq,axiom,
    ! [A4: set_o,K2: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_o @ A4 ) )
       => ( ( finite_card_list_o
            @ ( collect_list_o
              @ ^ [Xs: list_o] :
                  ( ( ( size_size_list_o @ Xs )
                    = K2 )
                  & ( distinct_o @ Xs )
                  & ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_o @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_o @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6622_card__lists__distinct__length__eq,axiom,
    ! [A4: set_int,K2: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_int @ A4 ) )
       => ( ( finite_card_list_int
            @ ( collect_list_int
              @ ^ [Xs: list_int] :
                  ( ( ( size_size_list_int @ Xs )
                    = K2 )
                  & ( distinct_int @ Xs )
                  & ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_int @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_int @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6623_card__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,K2: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ord_less_eq_nat @ K2 @ ( finite_card_nat @ A4 ) )
       => ( ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [Xs: list_nat] :
                  ( ( ( size_size_list_nat @ Xs )
                    = K2 )
                  & ( distinct_nat @ Xs )
                  & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 ) ) ) )
          = ( groups708209901874060359at_nat
            @ ^ [X4: nat] : X4
            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_nat @ A4 ) ) ) ) ) ) ).

% card_lists_distinct_length_eq
thf(fact_6624_gchoose__row__sum__weighted,axiom,
    ! [R3: complex,M2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R3 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R3 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M2 ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M2 ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R3 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6625_gchoose__row__sum__weighted,axiom,
    ! [R3: rat,M2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R3 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R3 @ ( 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 @ R3 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6626_gchoose__row__sum__weighted,axiom,
    ! [R3: real,M2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R3 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R3 @ ( 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 @ R3 @ ( suc @ M2 ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_6627_vebt__buildup_Opelims,axiom,
    ! [X: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va: nat] :
                  ( ( X
                    = ( suc @ ( suc @ Va ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) )
                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
                       => ( Y
                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ ( suc @ Va ) ) @ ( replicate_VEBT_VEBT @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_6628_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_6629_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_6630_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_6631_atMost__eq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( set_ord_atMost_nat @ X )
        = ( set_ord_atMost_nat @ Y ) )
      = ( X = Y ) ) ).

% atMost_eq_iff
thf(fact_6632_atMost__iff,axiom,
    ! [I2: real,K2: real] :
      ( ( member_real @ I2 @ ( set_ord_atMost_real @ K2 ) )
      = ( ord_less_eq_real @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6633_atMost__iff,axiom,
    ! [I2: set_nat,K2: set_nat] :
      ( ( member_set_nat @ I2 @ ( set_or4236626031148496127et_nat @ K2 ) )
      = ( ord_less_eq_set_nat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6634_atMost__iff,axiom,
    ! [I2: rat,K2: rat] :
      ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K2 ) )
      = ( ord_less_eq_rat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6635_atMost__iff,axiom,
    ! [I2: num,K2: num] :
      ( ( member_num @ I2 @ ( set_ord_atMost_num @ K2 ) )
      = ( ord_less_eq_num @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6636_atMost__iff,axiom,
    ! [I2: int,K2: int] :
      ( ( member_int @ I2 @ ( set_ord_atMost_int @ K2 ) )
      = ( ord_less_eq_int @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6637_atMost__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K2 ) )
      = ( ord_less_eq_nat @ I2 @ K2 ) ) ).

% atMost_iff
thf(fact_6638_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_6639_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_6640_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_6641_of__bool__less__eq__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P2
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_6642_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_6643_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_6644_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_6645_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_6646_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_6647_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_6648_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P2 )
        = zero_zero_complex )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6649_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P2 )
        = zero_zero_real )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6650_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P2 )
        = zero_zero_rat )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6651_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = zero_zero_nat )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6652_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = zero_zero_int )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6653_of__bool__eq__0__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = zero_z3403309356797280102nteger )
      = ~ P2 ) ).

% of_bool_eq_0_iff
thf(fact_6654_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_6655_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_6656_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_6657_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_6658_of__bool__less__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P2
        & Q ) ) ).

% of_bool_less_iff
thf(fact_6659_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P2 )
        = one_one_complex )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_6660_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P2 )
        = one_one_real )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_6661_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = one_one_nat )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_6662_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = one_one_int )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_6663_of__bool__eq__1__iff,axiom,
    ! [P2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = one_one_Code_integer )
      = P2 ) ).

% of_bool_eq_1_iff
thf(fact_6664_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_6665_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_6666_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_6667_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_6668_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_6669_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n3304061248610475627l_real @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6670_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6671_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6672_of__nat__of__bool,axiom,
    ! [P2: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( zero_n356916108424825756nteger @ P2 ) ) ).

% of_nat_of_bool
thf(fact_6673_of__bool__or__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P2
          | Q ) )
      = ( ord_max_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6674_of__bool__or__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P2
          | Q ) )
      = ( ord_max_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6675_of__bool__or__iff,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P2
          | Q ) )
      = ( ord_max_Code_integer @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_or_iff
thf(fact_6676_finite__atMost,axiom,
    ! [K2: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K2 ) ) ).

% finite_atMost
thf(fact_6677_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6678_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6679_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6680_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6681_zero__less__of__bool__iff,axiom,
    ! [P2: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P2 ) )
      = P2 ) ).

% zero_less_of_bool_iff
thf(fact_6682_atMost__subset__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_or4236626031148496127et_nat @ X ) @ ( set_or4236626031148496127et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6683_atMost__subset__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6684_atMost__subset__iff,axiom,
    ! [X: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6685_atMost__subset__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6686_atMost__subset__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X @ Y ) ) ).

% atMost_subset_iff
thf(fact_6687_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6688_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6689_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6690_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6691_of__bool__less__one__iff,axiom,
    ! [P2: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer )
      = ~ P2 ) ).

% of_bool_less_one_iff
thf(fact_6692_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n1201886186963655149omplex @ ~ P2 )
      = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_6693_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n3304061248610475627l_real @ ~ P2 )
      = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_6694_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P2 )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_6695_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n2684676970156552555ol_int @ ~ P2 )
      = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_6696_of__bool__not__iff,axiom,
    ! [P2: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P2 )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P2 ) ) ) ).

% of_bool_not_iff
thf(fact_6697_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_6698_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K2 ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_6699_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K2 ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_6700_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K2 ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_6701_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K2 ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_6702_gbinomial__0_I2_J,axiom,
    ! [K2: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K2 ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_6703_gbinomial__0_I1_J,axiom,
    ! [A: code_integer] :
      ( ( gbinom8545251970709558553nteger @ A @ zero_zero_nat )
      = one_one_Code_integer ) ).

% gbinomial_0(1)
thf(fact_6704_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_6705_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_6706_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_6707_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_6708_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_6709_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_6710_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_6711_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_6712_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_6713_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_6714_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_6715_sum_OatMost__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6716_sum_OatMost__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6717_sum_OatMost__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6718_sum_OatMost__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% sum.atMost_Suc
thf(fact_6719_prod_OatMost__Suc,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( groups129246275422532515t_real @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6720_prod_OatMost__Suc,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6721_prod_OatMost__Suc,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6722_prod_OatMost__Suc,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ N ) ) @ ( G2 @ ( suc @ N ) ) ) ) ).

% prod.atMost_Suc
thf(fact_6723_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_6724_distinct__swap,axiom,
    ! [I2: nat,Xs2: list_VEBT_VEBT,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( distinct_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs2 @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ J2 ) ) @ J2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) )
          = ( distinct_VEBT_VEBT @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_6725_distinct__swap,axiom,
    ! [I2: nat,Xs2: list_o,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs2 ) )
       => ( ( distinct_o @ ( list_update_o @ ( list_update_o @ Xs2 @ I2 @ ( nth_o @ Xs2 @ J2 ) ) @ J2 @ ( nth_o @ Xs2 @ I2 ) ) )
          = ( distinct_o @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_6726_distinct__swap,axiom,
    ! [I2: nat,Xs2: list_nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs2 @ I2 @ ( nth_nat @ Xs2 @ J2 ) ) @ J2 @ ( nth_nat @ Xs2 @ I2 ) ) )
          = ( distinct_nat @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_6727_distinct__swap,axiom,
    ! [I2: nat,Xs2: list_int,J2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs2 ) )
       => ( ( distinct_int @ ( list_update_int @ ( list_update_int @ Xs2 @ I2 @ ( nth_int @ Xs2 @ J2 ) ) @ J2 @ ( nth_int @ Xs2 @ I2 ) ) )
          = ( distinct_int @ Xs2 ) ) ) ) ).

% distinct_swap
thf(fact_6728_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: real > real,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( times_times_real @ ( F2 @ X4 ) @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups8097168146408367636l_real @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6729_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: int > real,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( times_times_real @ ( F2 @ X4 ) @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups8778361861064173332t_real @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6730_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: complex > real,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( times_times_real @ ( F2 @ X4 ) @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups5808333547571424918x_real @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6731_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: real > rat,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X4: real] : ( times_times_rat @ ( F2 @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups1300246762558778688al_rat @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6732_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: int > rat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( F2 @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6733_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: complex > rat,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X4: complex] : ( times_times_rat @ ( F2 @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups5058264527183730370ex_rat @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6734_sum__mult__of__bool__eq,axiom,
    ! [A4: set_nat,F2: nat > rat,P2: nat > $o] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( F2 @ X4 ) @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6735_sum__mult__of__bool__eq,axiom,
    ! [A4: set_real,F2: real > nat,P2: real > $o] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( times_times_nat @ ( F2 @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups1935376822645274424al_nat @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6736_sum__mult__of__bool__eq,axiom,
    ! [A4: set_int,F2: int > nat,P2: int > $o] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( F2 @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6737_sum__mult__of__bool__eq,axiom,
    ! [A4: set_complex,F2: complex > nat,P2: complex > $o] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( times_times_nat @ ( F2 @ X4 ) @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) )
          @ A4 )
        = ( groups5693394587270226106ex_nat @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_mult_of_bool_eq
thf(fact_6738_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: real > $o,F2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups8097168146408367636l_real
          @ ^ [X4: real] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups8097168146408367636l_real @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6739_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups8778361861064173332t_real
          @ ^ [X4: int] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups8778361861064173332t_real @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6740_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: complex > $o,F2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5808333547571424918x_real
          @ ^ [X4: complex] : ( times_times_real @ ( zero_n3304061248610475627l_real @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups5808333547571424918x_real @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6741_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: real > $o,F2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1300246762558778688al_rat
          @ ^ [X4: real] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups1300246762558778688al_rat @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6742_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups3906332499630173760nt_rat
          @ ^ [X4: int] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups3906332499630173760nt_rat @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6743_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: complex > $o,F2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5058264527183730370ex_rat
          @ ^ [X4: complex] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups5058264527183730370ex_rat @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6744_sum__of__bool__mult__eq,axiom,
    ! [A4: set_nat,P2: nat > $o,F2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [X4: nat] : ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups2906978787729119204at_rat @ F2 @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6745_sum__of__bool__mult__eq,axiom,
    ! [A4: set_real,P2: real > $o,F2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X4: real] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups1935376822645274424al_nat @ F2 @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6746_sum__of__bool__mult__eq,axiom,
    ! [A4: set_int,P2: int > $o,F2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X4: int] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups4541462559716669496nt_nat @ F2 @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6747_sum__of__bool__mult__eq,axiom,
    ! [A4: set_complex,P2: complex > $o,F2: complex > nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X4: complex] : ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( P2 @ X4 ) ) @ ( F2 @ X4 ) )
          @ A4 )
        = ( groups5693394587270226106ex_nat @ F2 @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) ) ) ).

% sum_of_bool_mult_eq
thf(fact_6748_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_6749_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_6750_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_6751_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs: list_complex] :
              ( ( ( size_s3451745648224563538omplex @ Xs )
                = N )
              & ( distinct_complex @ Xs )
              & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6752_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs: list_VEBT_VEBT] :
              ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                = N )
              & ( distinct_VEBT_VEBT @ Xs )
              & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6753_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs: list_o] :
              ( ( ( size_size_list_o @ Xs )
                = N )
              & ( distinct_o @ Xs )
              & ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6754_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs: list_int] :
              ( ( ( size_size_list_int @ Xs )
                = N )
              & ( distinct_int @ Xs )
              & ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6755_finite__lists__distinct__length__eq,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs: list_nat] :
              ( ( ( size_size_list_nat @ Xs )
                = N )
              & ( distinct_nat @ Xs )
              & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 ) ) ) ) ) ).

% finite_lists_distinct_length_eq
thf(fact_6756_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_6757_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_6758_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_6759_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_6760_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_6761_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_6762_one__mod__2__pow__eq,axiom,
    ! [N: nat] :
      ( ( modulo8411746178871703098atural @ one_one_Code_natural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( zero_n8403883297036319079atural @ ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% one_mod_2_pow_eq
thf(fact_6763_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_6764_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_6765_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_6766_of__bool__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = ( zero_n2687167440665602831ol_nat @ Q2 ) )
      = ( P = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6767_of__bool__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = ( zero_n2684676970156552555ol_int @ Q2 ) )
      = ( P = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6768_of__bool__eq__iff,axiom,
    ! [P: $o,Q2: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = ( zero_n356916108424825756nteger @ Q2 ) )
      = ( P = Q2 ) ) ).

% of_bool_eq_iff
thf(fact_6769_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n3304061248610475627l_real
        @ ( P2
          & Q ) )
      = ( times_times_real @ ( zero_n3304061248610475627l_real @ P2 ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).

% of_bool_conj
thf(fact_6770_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P2
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_6771_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2687167440665602831ol_nat
        @ ( P2
          & Q ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).

% of_bool_conj
thf(fact_6772_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n2684676970156552555ol_int
        @ ( P2
          & Q ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).

% of_bool_conj
thf(fact_6773_of__bool__conj,axiom,
    ! [P2: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P2
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P2 ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_6774_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_6775_not__empty__eq__Iic__eq__empty,axiom,
    ! [H: real] :
      ( bot_bot_set_real
     != ( set_ord_atMost_real @ H ) ) ).

% not_empty_eq_Iic_eq_empty
thf(fact_6776_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_6777_infinite__Iic,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_atMost_int @ A ) ) ).

% infinite_Iic
thf(fact_6778_not__Iic__eq__Icc,axiom,
    ! [H3: int,L: int,H: int] :
      ( ( set_ord_atMost_int @ H3 )
     != ( set_or1266510415728281911st_int @ L @ H ) ) ).

% not_Iic_eq_Icc
thf(fact_6779_not__Iic__eq__Icc,axiom,
    ! [H3: real,L: real,H: real] :
      ( ( set_ord_atMost_real @ H3 )
     != ( set_or1222579329274155063t_real @ L @ H ) ) ).

% not_Iic_eq_Icc
thf(fact_6780_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X4: real] : ( ord_less_eq_real @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6781_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_6782_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X4: rat] : ( ord_less_eq_rat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6783_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X4: num] : ( ord_less_eq_num @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6784_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X4: int] : ( ord_less_eq_int @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6785_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X4: nat] : ( ord_less_eq_nat @ X4 @ U2 ) ) ) ) ).

% atMost_def
thf(fact_6786_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6787_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6788_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6789_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6790_zero__less__eq__of__bool,axiom,
    ! [P2: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P2 ) ) ).

% zero_less_eq_of_bool
thf(fact_6791_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P2 ) @ one_one_real ) ).

% of_bool_less_eq_one
thf(fact_6792_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_6793_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P2 ) @ one_one_nat ) ).

% of_bool_less_eq_one
thf(fact_6794_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P2 ) @ one_one_int ) ).

% of_bool_less_eq_one
thf(fact_6795_of__bool__less__eq__one,axiom,
    ! [P2: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P2 ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_6796_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_6797_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_6798_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_6799_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_6800_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_6801_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_6802_split__of__bool,axiom,
    ! [P2: complex > $o,P: $o] :
      ( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
      = ( ( P
         => ( P2 @ one_one_complex ) )
        & ( ~ P
         => ( P2 @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_6803_split__of__bool,axiom,
    ! [P2: real > $o,P: $o] :
      ( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
      = ( ( P
         => ( P2 @ one_one_real ) )
        & ( ~ P
         => ( P2 @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_6804_split__of__bool,axiom,
    ! [P2: rat > $o,P: $o] :
      ( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( ( P
         => ( P2 @ one_one_rat ) )
        & ( ~ P
         => ( P2 @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_6805_split__of__bool,axiom,
    ! [P2: nat > $o,P: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( ( P
         => ( P2 @ one_one_nat ) )
        & ( ~ P
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_6806_split__of__bool,axiom,
    ! [P2: int > $o,P: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( ( P
         => ( P2 @ one_one_int ) )
        & ( ~ P
         => ( P2 @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_6807_split__of__bool,axiom,
    ! [P2: code_integer > $o,P: $o] :
      ( ( P2 @ ( zero_n356916108424825756nteger @ P ) )
      = ( ( P
         => ( P2 @ one_one_Code_integer ) )
        & ( ~ P
         => ( P2 @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_6808_split__of__bool__asm,axiom,
    ! [P2: complex > $o,P: $o] :
      ( ( P2 @ ( zero_n1201886186963655149omplex @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_complex ) )
            | ( ~ P
              & ~ ( P2 @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6809_split__of__bool__asm,axiom,
    ! [P2: real > $o,P: $o] :
      ( ( P2 @ ( zero_n3304061248610475627l_real @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_real ) )
            | ( ~ P
              & ~ ( P2 @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6810_split__of__bool__asm,axiom,
    ! [P2: rat > $o,P: $o] :
      ( ( P2 @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_rat ) )
            | ( ~ P
              & ~ ( P2 @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6811_split__of__bool__asm,axiom,
    ! [P2: nat > $o,P: $o] :
      ( ( P2 @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_nat ) )
            | ( ~ P
              & ~ ( P2 @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6812_split__of__bool__asm,axiom,
    ! [P2: int > $o,P: $o] :
      ( ( P2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_int ) )
            | ( ~ P
              & ~ ( P2 @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6813_split__of__bool__asm,axiom,
    ! [P2: code_integer > $o,P: $o] :
      ( ( P2 @ ( zero_n356916108424825756nteger @ P ) )
      = ( ~ ( ( P
              & ~ ( P2 @ one_one_Code_integer ) )
            | ( ~ P
              & ~ ( P2 @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_6814_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_6815_lessThan__Suc__atMost,axiom,
    ! [K2: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K2 ) )
      = ( set_ord_atMost_nat @ K2 ) ) ).

% lessThan_Suc_atMost
thf(fact_6816_atMost__Suc,axiom,
    ! [K2: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K2 ) )
      = ( insert_nat @ ( suc @ K2 ) @ ( set_ord_atMost_nat @ K2 ) ) ) ).

% atMost_Suc
thf(fact_6817_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_6818_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_6819_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_VEBT_VEBT,I2: nat,J2: nat] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( ord_less_nat @ J2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
         => ( ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
              = ( nth_VEBT_VEBT @ Xs2 @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_6820_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_o,I2: nat,J2: nat] :
      ( ( distinct_o @ Xs2 )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_o @ Xs2 ) )
         => ( ( ( nth_o @ Xs2 @ I2 )
              = ( nth_o @ Xs2 @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_6821_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_nat,I2: nat,J2: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
         => ( ( ( nth_nat @ Xs2 @ I2 )
              = ( nth_nat @ Xs2 @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_6822_nth__eq__iff__index__eq,axiom,
    ! [Xs2: list_int,I2: nat,J2: nat] :
      ( ( distinct_int @ Xs2 )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_int @ Xs2 ) )
         => ( ( ( nth_int @ Xs2 @ I2 )
              = ( nth_int @ Xs2 @ J2 ) )
            = ( I2 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_6823_distinct__conv__nth,axiom,
    ( distinct_VEBT_VEBT
    = ( ^ [Xs: list_VEBT_VEBT] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
             => ( ( I != J )
               => ( ( nth_VEBT_VEBT @ Xs @ I )
                 != ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_6824_distinct__conv__nth,axiom,
    ( distinct_o
    = ( ^ [Xs: list_o] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
             => ( ( I != J )
               => ( ( nth_o @ Xs @ I )
                 != ( nth_o @ Xs @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_6825_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs: list_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
             => ( ( I != J )
               => ( ( nth_nat @ Xs @ I )
                 != ( nth_nat @ Xs @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_6826_distinct__conv__nth,axiom,
    ( distinct_int
    = ( ^ [Xs: list_int] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
             => ( ( I != J )
               => ( ( nth_int @ Xs @ I )
                 != ( nth_int @ Xs @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_6827_card__distinct,axiom,
    ! [Xs2: list_complex] :
      ( ( ( finite_card_complex @ ( set_complex2 @ Xs2 ) )
        = ( size_s3451745648224563538omplex @ Xs2 ) )
     => ( distinct_complex @ Xs2 ) ) ).

% card_distinct
thf(fact_6828_card__distinct,axiom,
    ! [Xs2: list_list_nat] :
      ( ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
        = ( size_s3023201423986296836st_nat @ Xs2 ) )
     => ( distinct_list_nat @ Xs2 ) ) ).

% card_distinct
thf(fact_6829_card__distinct,axiom,
    ! [Xs2: list_set_nat] :
      ( ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs2 ) )
        = ( size_s3254054031482475050et_nat @ Xs2 ) )
     => ( distinct_set_nat @ Xs2 ) ) ).

% card_distinct
thf(fact_6830_card__distinct,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) )
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) )
     => ( distinct_VEBT_VEBT @ Xs2 ) ) ).

% card_distinct
thf(fact_6831_card__distinct,axiom,
    ! [Xs2: list_o] :
      ( ( ( finite_card_o @ ( set_o2 @ Xs2 ) )
        = ( size_size_list_o @ Xs2 ) )
     => ( distinct_o @ Xs2 ) ) ).

% card_distinct
thf(fact_6832_card__distinct,axiom,
    ! [Xs2: list_nat] :
      ( ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
        = ( size_size_list_nat @ Xs2 ) )
     => ( distinct_nat @ Xs2 ) ) ).

% card_distinct
thf(fact_6833_card__distinct,axiom,
    ! [Xs2: list_int] :
      ( ( ( finite_card_int @ ( set_int2 @ Xs2 ) )
        = ( size_size_list_int @ Xs2 ) )
     => ( distinct_int @ Xs2 ) ) ).

% card_distinct
thf(fact_6834_distinct__card,axiom,
    ! [Xs2: list_complex] :
      ( ( distinct_complex @ Xs2 )
     => ( ( finite_card_complex @ ( set_complex2 @ Xs2 ) )
        = ( size_s3451745648224563538omplex @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6835_distinct__card,axiom,
    ! [Xs2: list_list_nat] :
      ( ( distinct_list_nat @ Xs2 )
     => ( ( finite_card_list_nat @ ( set_list_nat2 @ Xs2 ) )
        = ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6836_distinct__card,axiom,
    ! [Xs2: list_set_nat] :
      ( ( distinct_set_nat @ Xs2 )
     => ( ( finite_card_set_nat @ ( set_set_nat2 @ Xs2 ) )
        = ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6837_distinct__card,axiom,
    ! [Xs2: list_VEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( finite7802652506058667612T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) )
        = ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6838_distinct__card,axiom,
    ! [Xs2: list_o] :
      ( ( distinct_o @ Xs2 )
     => ( ( finite_card_o @ ( set_o2 @ Xs2 ) )
        = ( size_size_list_o @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6839_distinct__card,axiom,
    ! [Xs2: list_nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( finite_card_nat @ ( set_nat2 @ Xs2 ) )
        = ( size_size_list_nat @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6840_distinct__card,axiom,
    ! [Xs2: list_int] :
      ( ( distinct_int @ Xs2 )
     => ( ( finite_card_int @ ( set_int2 @ Xs2 ) )
        = ( size_size_list_int @ Xs2 ) ) ) ).

% distinct_card
thf(fact_6841_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_6842_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_6843_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_6844_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_6845_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_6846_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_6847_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_6848_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_6849_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_6850_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_6851_distinct__Ex1,axiom,
    ! [Xs2: list_complex,X: complex] :
      ( ( distinct_complex @ Xs2 )
     => ( ( member_complex @ X @ ( set_complex2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s3451745648224563538omplex @ Xs2 ) )
            & ( ( nth_complex @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s3451745648224563538omplex @ Xs2 ) )
                  & ( ( nth_complex @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6852_distinct__Ex1,axiom,
    ! [Xs2: list_real,X: real] :
      ( ( distinct_real @ Xs2 )
     => ( ( member_real @ X @ ( set_real2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_real @ Xs2 ) )
            & ( ( nth_real @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_real @ Xs2 ) )
                  & ( ( nth_real @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6853_distinct__Ex1,axiom,
    ! [Xs2: list_set_nat,X: set_nat] :
      ( ( distinct_set_nat @ Xs2 )
     => ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
            & ( ( nth_set_nat @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s3254054031482475050et_nat @ Xs2 ) )
                  & ( ( nth_set_nat @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6854_distinct__Ex1,axiom,
    ! [Xs2: list_VEBT_VEBT,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
            & ( ( nth_VEBT_VEBT @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
                  & ( ( nth_VEBT_VEBT @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6855_distinct__Ex1,axiom,
    ! [Xs2: list_o,X: $o] :
      ( ( distinct_o @ Xs2 )
     => ( ( member_o @ X @ ( set_o2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_o @ Xs2 ) )
            & ( ( nth_o @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_o @ Xs2 ) )
                  & ( ( nth_o @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6856_distinct__Ex1,axiom,
    ! [Xs2: list_nat,X: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs2 ) )
            & ( ( nth_nat @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs2 ) )
                  & ( ( nth_nat @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6857_distinct__Ex1,axiom,
    ! [Xs2: list_int,X: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( member_int @ X @ ( set_int2 @ Xs2 ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_int @ Xs2 ) )
            & ( ( nth_int @ Xs2 @ X3 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_int @ Xs2 ) )
                  & ( ( nth_int @ Xs2 @ Y6 )
                    = X ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_6858_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X @ 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 @ X @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6859_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: rat,X: 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 @ X @ 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 @ X @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6860_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M2: nat,A: real,X: 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 @ X @ 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 @ X @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_6861_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_6862_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_6863_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_6864_gbinomial__absorb__comp,axiom,
    ! [A: complex,K2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( gbinomial_complex @ A @ K2 ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_6865_gbinomial__absorb__comp,axiom,
    ! [A: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ A @ K2 ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_6866_gbinomial__absorb__comp,axiom,
    ! [A: real,K2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( gbinomial_real @ A @ K2 ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K2 ) ) ) ).

% gbinomial_absorb_comp
thf(fact_6867_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_6868_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_6869_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_6870_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_6871_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_6872_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_6873_sum_OatMost__Suc__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_rat @ ( G2 @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6874_sum_OatMost__Suc__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_int @ ( G2 @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6875_sum_OatMost__Suc__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_nat @ ( G2 @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6876_sum_OatMost__Suc__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( plus_plus_real @ ( G2 @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_6877_sum__telescope,axiom,
    ! [F2: nat > rat,I2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( minus_minus_rat @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_rat @ ( F2 @ zero_zero_nat ) @ ( F2 @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_6878_sum__telescope,axiom,
    ! [F2: nat > int,I2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( minus_minus_int @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_int @ ( F2 @ zero_zero_nat ) @ ( F2 @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_6879_sum__telescope,axiom,
    ! [F2: nat > real,I2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( minus_minus_real @ ( F2 @ I ) @ ( F2 @ ( suc @ I ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_real @ ( F2 @ zero_zero_nat ) @ ( F2 @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_6880_polyfun__eq__coeffs,axiom,
    ! [C2: nat > complex,N: nat,D: nat > complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups2073611262835488442omplex
              @ ^ [I: nat] : ( times_times_complex @ ( D @ I ) @ ( power_power_complex @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
           => ( ( C2 @ I )
              = ( D @ I ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6881_polyfun__eq__coeffs,axiom,
    ! [C2: nat > real,N: nat,D: nat > real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( D @ I ) @ ( power_power_real @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
           => ( ( C2 @ I )
              = ( D @ I ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_6882_prod_OatMost__Suc__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_real @ ( G2 @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6883_prod_OatMost__Suc__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_rat @ ( G2 @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6884_prod_OatMost__Suc__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_int @ ( G2 @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6885_prod_OatMost__Suc__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ N ) ) )
      = ( times_times_nat @ ( G2 @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% prod.atMost_Suc_shift
thf(fact_6886_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( groups3542108847815614940at_nat @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_6887_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( groups6591440286371151544t_real @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups6591440286371151544t_real
        @ ^ [J: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% sum.nested_swap'
thf(fact_6888_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_6889_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_6890_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_6891_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_6892_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_6893_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_6894_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_6895_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_6896_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > int,N: nat] :
      ( ( groups705719431365010083at_int
        @ ^ [I: nat] : ( groups705719431365010083at_int @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups705719431365010083at_int
        @ ^ [J: nat] :
            ( groups705719431365010083at_int
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_6897_prod_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( groups708209901874060359at_nat @ ( A @ I ) @ ( set_ord_lessThan_nat @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( groups708209901874060359at_nat
        @ ^ [J: nat] :
            ( groups708209901874060359at_nat
            @ ^ [I: nat] : ( A @ I @ J )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
        @ ( set_ord_lessThan_nat @ N ) ) ) ).

% prod.nested_swap'
thf(fact_6898_sum__choose__lower,axiom,
    ! [R3: nat,N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N ) ) @ N ) ) ).

% sum_choose_lower
thf(fact_6899_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_6900_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_6901_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_6902_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_6903_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_6904_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_6905_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_6906_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_6907_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: nat > complex,N: nat,K2: nat] :
      ( ! [W: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ W @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( C2 @ K2 )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6908_zero__polynom__imp__zero__coeffs,axiom,
    ! [C2: nat > real,N: nat,K2: nat] :
      ( ! [W: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ W @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( C2 @ K2 )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_6909_polyfun__eq__0,axiom,
    ! [C2: nat > complex,N: nat] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_complex ) )
      = ( ! [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
           => ( ( C2 @ I )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_6910_polyfun__eq__0,axiom,
    ! [C2: nat > real,N: nat] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = zero_zero_real ) )
      = ( ! [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
           => ( ( C2 @ I )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_6911_sum_OatMost__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat @ ( G2 @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_6912_sum_OatMost__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int @ ( G2 @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_6913_sum_OatMost__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_nat @ ( G2 @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_6914_sum_OatMost__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real @ ( G2 @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% sum.atMost_shift
thf(fact_6915_sum__up__index__split,axiom,
    ! [F2: nat > rat,M2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat @ F2 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F2 @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups2906978787729119204at_rat @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6916_sum__up__index__split,axiom,
    ! [F2: nat > int,M2: nat,N: nat] :
      ( ( groups3539618377306564664at_int @ F2 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F2 @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3539618377306564664at_int @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6917_sum__up__index__split,axiom,
    ! [F2: nat > nat,M2: nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ F2 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F2 @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups3542108847815614940at_nat @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6918_sum__up__index__split,axiom,
    ! [F2: nat > real,M2: nat,N: nat] :
      ( ( groups6591440286371151544t_real @ F2 @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F2 @ ( set_ord_atMost_nat @ M2 ) ) @ ( groups6591440286371151544t_real @ F2 @ ( set_or1269000886237332187st_nat @ ( suc @ M2 ) @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ).

% sum_up_index_split
thf(fact_6919_prod_OatMost__shift,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_real @ ( G2 @ zero_zero_nat )
        @ ( groups129246275422532515t_real
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_6920_prod_OatMost__shift,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_rat @ ( G2 @ zero_zero_nat )
        @ ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_6921_prod_OatMost__shift,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_int @ ( G2 @ zero_zero_nat )
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_6922_prod_OatMost__shift,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ ( G2 @ zero_zero_nat )
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( G2 @ ( suc @ I ) )
          @ ( set_ord_lessThan_nat @ N ) ) ) ) ).

% prod.atMost_shift
thf(fact_6923_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6924_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6925_gbinomial__r__part__sum,axiom,
    ! [M2: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% gbinomial_r_part_sum
thf(fact_6926_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_6927_vandermonde,axiom,
    ! [M2: nat,N: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M2 @ N ) @ R3 ) ) ).

% vandermonde
thf(fact_6928_bits__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [A3: nat] :
          ( ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: nat,B3: $o] :
            ( ( P2 @ 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 )
             => ( P2 @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B3 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6929_bits__induct,axiom,
    ! [P2: int > $o,A: int] :
      ( ! [A3: int] :
          ( ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: int,B3: $o] :
            ( ( P2 @ 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 )
             => ( P2 @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B3 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6930_bits__induct,axiom,
    ! [P2: code_integer > $o,A: code_integer] :
      ( ! [A3: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A3 )
         => ( P2 @ A3 ) )
     => ( ! [A3: code_integer,B3: $o] :
            ( ( P2 @ A3 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A3 )
             => ( P2 @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B3 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) ) ) ) )
       => ( P2 @ A ) ) ) ).

% bits_induct
thf(fact_6931_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_6932_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_6933_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_6934_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_6935_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_6936_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_6937_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_6938_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_6939_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_6940_sum__gp__basic,axiom,
    ! [X: code_integer,N: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ one_one_Code_integer @ X ) @ ( groups7501900531339628137nteger @ ( power_8256067586552552935nteger @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_6941_sum__gp__basic,axiom,
    ! [X: complex,N: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_6942_sum__gp__basic,axiom,
    ! [X: rat,N: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_6943_sum__gp__basic,axiom,
    ! [X: int,N: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_6944_sum__gp__basic,axiom,
    ! [X: real,N: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ).

% sum_gp_basic
thf(fact_6945_polyfun__finite__roots,axiom,
    ! [C2: nat > complex,N: nat] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X4: complex] :
              ( ( groups2073611262835488442omplex
                @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ X4 @ I ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_complex ) ) )
      = ( ? [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
            & ( ( C2 @ I )
             != zero_zero_complex ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6946_polyfun__finite__roots,axiom,
    ! [C2: nat > real,N: nat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X4: real] :
              ( ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ X4 @ I ) )
                @ ( set_ord_atMost_nat @ N ) )
              = zero_zero_real ) ) )
      = ( ? [I: nat] :
            ( ( ord_less_eq_nat @ I @ N )
            & ( ( C2 @ I )
             != zero_zero_real ) ) ) ) ).

% polyfun_finite_roots
thf(fact_6947_polyfun__roots__finite,axiom,
    ! [C2: nat > complex,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z4 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_complex ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6948_polyfun__roots__finite,axiom,
    ! [C2: nat > real,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Z4: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z4 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = zero_zero_real ) ) ) ) ) ).

% polyfun_roots_finite
thf(fact_6949_polyfun__roots__card,axiom,
    ! [C2: nat > complex,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ord_less_eq_nat
          @ ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z4: complex] :
                  ( ( groups2073611262835488442omplex
                    @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z4 @ I ) )
                    @ ( set_ord_atMost_nat @ N ) )
                  = zero_zero_complex ) ) )
          @ N ) ) ) ).

% polyfun_roots_card
thf(fact_6950_polyfun__roots__card,axiom,
    ! [C2: nat > real,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ord_less_eq_nat
          @ ( finite_card_real
            @ ( collect_real
              @ ^ [Z4: real] :
                  ( ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z4 @ I ) )
                    @ ( set_ord_atMost_nat @ N ) )
                  = zero_zero_real ) ) )
          @ N ) ) ) ).

% polyfun_roots_card
thf(fact_6951_polyfun__linear__factor__root,axiom,
    ! [C2: nat > complex,A: complex,N: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex )
     => ~ ! [B3: nat > complex] :
            ~ ! [Z5: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z5 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I: nat] : ( times_times_complex @ ( B3 @ I ) @ ( power_power_complex @ Z5 @ I ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6952_polyfun__linear__factor__root,axiom,
    ! [C2: nat > rat,A: rat,N: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat )
     => ~ ! [B3: nat > rat] :
            ~ ! [Z5: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ Z5 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I: nat] : ( times_times_rat @ ( B3 @ I ) @ ( power_power_rat @ Z5 @ I ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6953_polyfun__linear__factor__root,axiom,
    ! [C2: nat > int,A: int,N: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( C2 @ I ) @ ( power_power_int @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int )
     => ~ ! [B3: nat > int] :
            ~ ! [Z5: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I: nat] : ( times_times_int @ ( C2 @ I ) @ ( power_power_int @ Z5 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_int @ ( minus_minus_int @ Z5 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I: nat] : ( times_times_int @ ( B3 @ I ) @ ( power_power_int @ Z5 @ I ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6954_polyfun__linear__factor__root,axiom,
    ! [C2: nat > real,A: real,N: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real )
     => ~ ! [B3: nat > real] :
            ~ ! [Z5: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z5 @ I ) )
                  @ ( set_ord_atMost_nat @ N ) )
                = ( times_times_real @ ( minus_minus_real @ Z5 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( B3 @ I ) @ ( power_power_real @ Z5 @ I ) )
                    @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_6955_polyfun__linear__factor,axiom,
    ! [C2: nat > complex,N: nat,A: complex] :
    ? [B3: nat > complex] :
    ! [Z5: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z5 @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z5 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( B3 @ I ) @ ( power_power_complex @ Z5 @ I ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6956_polyfun__linear__factor,axiom,
    ! [C2: nat > rat,N: nat,A: rat] :
    ? [B3: nat > rat] :
    ! [Z5: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ Z5 @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z5 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( B3 @ I ) @ ( power_power_rat @ Z5 @ I ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( C2 @ I ) @ ( power_power_rat @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6957_polyfun__linear__factor,axiom,
    ! [C2: nat > int,N: nat,A: int] :
    ? [B3: nat > int] :
    ! [Z5: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( times_times_int @ ( C2 @ I ) @ ( power_power_int @ Z5 @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z5 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( B3 @ I ) @ ( power_power_int @ Z5 @ I ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( C2 @ I ) @ ( power_power_int @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6958_polyfun__linear__factor,axiom,
    ! [C2: nat > real,N: nat,A: real] :
    ? [B3: nat > real] :
    ! [Z5: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z5 @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z5 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( B3 @ I ) @ ( power_power_real @ Z5 @ I ) )
            @ ( set_ord_lessThan_nat @ N ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ A @ I ) )
          @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% polyfun_linear_factor
thf(fact_6959_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: complex] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_complex @ ( power_power_complex @ X @ M2 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6960_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: rat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_rat @ ( power_power_rat @ X @ M2 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6961_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: int] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_int @ ( power_power_int @ X @ M2 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6962_sum__power__shift,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_or1269000886237332187st_nat @ M2 @ N ) )
        = ( times_times_real @ ( power_power_real @ X @ M2 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N @ M2 ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_6963_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_6964_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_6965_exp__mod__exp,axiom,
    ! [M2: nat,N: nat] :
      ( ( modulo8411746178871703098atural @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ N ) )
      = ( times_2397367101498566445atural @ ( zero_n8403883297036319079atural @ ( ord_less_nat @ M2 @ N ) ) @ ( power_7079662738309270450atural @ ( numera5444537566228673987atural @ ( bit0 @ one ) ) @ M2 ) ) ) ).

% exp_mod_exp
thf(fact_6966_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_6967_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_6968_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_6969_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_6970_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_6971_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_6972_sum_Oin__pairs__0,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups2906978787729119204at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I: nat] : ( plus_plus_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_6973_sum_Oin__pairs__0,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups3539618377306564664at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I: nat] : ( plus_plus_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_6974_sum_Oin__pairs__0,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups3542108847815614940at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( plus_plus_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_6975_sum_Oin__pairs__0,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups6591440286371151544t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( plus_plus_real @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% sum.in_pairs_0
thf(fact_6976_polyfun__rootbound,axiom,
    ! [C2: nat > complex,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( finite3207457112153483333omplex
            @ ( collect_complex
              @ ^ [Z4: complex] :
                  ( ( groups2073611262835488442omplex
                    @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z4 @ I ) )
                    @ ( set_ord_atMost_nat @ N ) )
                  = zero_zero_complex ) ) )
          & ( ord_less_eq_nat
            @ ( finite_card_complex
              @ ( collect_complex
                @ ^ [Z4: complex] :
                    ( ( groups2073611262835488442omplex
                      @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z4 @ I ) )
                      @ ( set_ord_atMost_nat @ N ) )
                    = zero_zero_complex ) ) )
            @ N ) ) ) ) ).

% polyfun_rootbound
thf(fact_6977_polyfun__rootbound,axiom,
    ! [C2: nat > real,K2: nat,N: nat] :
      ( ( ( C2 @ K2 )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ K2 @ N )
       => ( ( finite_finite_real
            @ ( collect_real
              @ ^ [Z4: real] :
                  ( ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z4 @ I ) )
                    @ ( set_ord_atMost_nat @ N ) )
                  = zero_zero_real ) ) )
          & ( ord_less_eq_nat
            @ ( finite_card_real
              @ ( collect_real
                @ ^ [Z4: real] :
                    ( ( groups6591440286371151544t_real
                      @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z4 @ I ) )
                      @ ( set_ord_atMost_nat @ N ) )
                    = zero_zero_real ) ) )
            @ N ) ) ) ) ).

% polyfun_rootbound
thf(fact_6978_polynomial__product,axiom,
    ! [M2: nat,A: nat > complex,N: nat,B: nat > complex,X: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_complex ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ X @ I ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J: nat] : ( times_times_complex @ ( B @ J ) @ ( power_power_complex @ X @ J ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_complex @ X @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6979_polynomial__product,axiom,
    ! [M2: nat,A: nat > rat,N: nat,B: nat > rat,X: rat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_rat ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ X @ I ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J: nat] : ( times_times_rat @ ( B @ J ) @ ( power_power_rat @ X @ J ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_rat @ X @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6980_polynomial__product,axiom,
    ! [M2: nat,A: nat > int,N: nat,B: nat > int,X: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_int ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ X @ I ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J: nat] : ( times_times_int @ ( B @ J ) @ ( power_power_int @ X @ J ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_int @ X @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6981_polynomial__product,axiom,
    ! [M2: nat,A: nat > real,N: nat,B: nat > real,X: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_real ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ X @ I ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J: nat] : ( times_times_real @ ( B @ J ) @ ( power_power_real @ X @ J ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_real @ X @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product
thf(fact_6982_prod_Oin__pairs__0,axiom,
    ! [G2: nat > real,N: nat] :
      ( ( groups129246275422532515t_real @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups129246275422532515t_real
        @ ^ [I: nat] : ( times_times_real @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_6983_prod_Oin__pairs__0,axiom,
    ! [G2: nat > rat,N: nat] :
      ( ( groups73079841787564623at_rat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [I: nat] : ( times_times_rat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_6984_prod_Oin__pairs__0,axiom,
    ! [G2: nat > int,N: nat] :
      ( ( groups705719431365010083at_int @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups705719431365010083at_int
        @ ^ [I: nat] : ( times_times_int @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_6985_prod_Oin__pairs__0,axiom,
    ! [G2: nat > nat,N: nat] :
      ( ( groups708209901874060359at_nat @ G2 @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
      = ( groups708209901874060359at_nat
        @ ^ [I: nat] : ( times_times_nat @ ( G2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) @ ( G2 @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I ) ) ) )
        @ ( set_ord_atMost_nat @ N ) ) ) ).

% prod.in_pairs_0
thf(fact_6986_polyfun__eq__const,axiom,
    ! [C2: nat > complex,N: nat,K2: complex] :
      ( ( ! [X4: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C2 @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C2 @ X4 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_6987_polyfun__eq__const,axiom,
    ! [C2: nat > real,N: nat,K2: real] :
      ( ( ! [X4: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ X4 @ I ) )
              @ ( set_ord_atMost_nat @ N ) )
            = K2 ) )
      = ( ( ( C2 @ zero_zero_nat )
          = K2 )
        & ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or1269000886237332187st_nat @ one_one_nat @ N ) )
           => ( ( C2 @ X4 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_6988_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_6989_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_6990_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_6991_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_6992_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_6993_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_6994_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_6995_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_6996_polynomial__product__nat,axiom,
    ! [M2: nat,A: nat > nat,N: nat,B: nat > nat,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M2 @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J3: nat] :
            ( ( ord_less_nat @ N @ J3 )
           => ( ( B @ J3 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( power_power_nat @ X @ I ) )
              @ ( set_ord_atMost_nat @ M2 ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J: nat] : ( times_times_nat @ ( B @ J ) @ ( power_power_nat @ X @ J ) )
              @ ( set_ord_atMost_nat @ N ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R ) )
                @ ( power_power_nat @ X @ R ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M2 @ N ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_6997_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_6998_set__update__distinct,axiom,
    ! [Xs2: list_VEBT_VEBT,N: nat,X: vEBT_VEBT] :
      ( ( distinct_VEBT_VEBT @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
       => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs2 @ N @ X ) )
          = ( insert_VEBT_VEBT @ X @ ( minus_5127226145743854075T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ ( insert_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs2 @ N ) @ bot_bo8194388402131092736T_VEBT ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_6999_set__update__distinct,axiom,
    ! [Xs2: list_o,N: nat,X: $o] :
      ( ( distinct_o @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_o @ Xs2 ) )
       => ( ( set_o2 @ ( list_update_o @ Xs2 @ N @ X ) )
          = ( insert_o @ X @ ( minus_minus_set_o @ ( set_o2 @ Xs2 ) @ ( insert_o @ ( nth_o @ Xs2 @ N ) @ bot_bot_set_o ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_7000_set__update__distinct,axiom,
    ! [Xs2: list_int,N: nat,X: int] :
      ( ( distinct_int @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_int @ Xs2 ) )
       => ( ( set_int2 @ ( list_update_int @ Xs2 @ N @ X ) )
          = ( insert_int @ X @ ( minus_minus_set_int @ ( set_int2 @ Xs2 ) @ ( insert_int @ ( nth_int @ Xs2 @ N ) @ bot_bot_set_int ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_7001_set__update__distinct,axiom,
    ! [Xs2: list_real,N: nat,X: real] :
      ( ( distinct_real @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_real @ Xs2 ) )
       => ( ( set_real2 @ ( list_update_real @ Xs2 @ N @ X ) )
          = ( insert_real @ X @ ( minus_minus_set_real @ ( set_real2 @ Xs2 ) @ ( insert_real @ ( nth_real @ Xs2 @ N ) @ bot_bot_set_real ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_7002_set__update__distinct,axiom,
    ! [Xs2: list_P6011104703257516679at_nat,N: nat,X: product_prod_nat_nat] :
      ( ( distin6923225563576452346at_nat @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs2 ) )
       => ( ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs2 @ N @ X ) )
          = ( insert8211810215607154385at_nat @ X @ ( minus_1356011639430497352at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ ( insert8211810215607154385at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs2 @ N ) @ bot_bo2099793752762293965at_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_7003_set__update__distinct,axiom,
    ! [Xs2: list_nat,N: nat,X: nat] :
      ( ( distinct_nat @ Xs2 )
     => ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs2 ) )
       => ( ( set_nat2 @ ( list_update_nat @ Xs2 @ N @ X ) )
          = ( insert_nat @ X @ ( minus_minus_set_nat @ ( set_nat2 @ Xs2 ) @ ( insert_nat @ ( nth_nat @ Xs2 @ N ) @ bot_bot_set_nat ) ) ) ) ) ) ).

% set_update_distinct
thf(fact_7004_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > complex,H: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups2073611262835488442omplex
            @ ^ [J: nat] : ( if_complex @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_complex @ ( J = K2 ) @ zero_zero_complex @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups2073611262835488442omplex
            @ ^ [J: nat] : ( if_complex @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7005_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_rat @ ( J = K2 ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J: nat] : ( if_rat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7006_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups3539618377306564664at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_int @ ( J = K2 ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups3539618377306564664at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7007_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_nat @ ( J = K2 ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7008_sum_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > real,H: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups6591440286371151544t_real
            @ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_real @ ( J = K2 ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups6591440286371151544t_real
            @ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_7009_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > code_integer,H: nat > code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups3455450783089532116nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_Code_integer @ ( J = K2 ) @ one_one_Code_integer @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups3455450783089532116nteger
            @ ^ [J: nat] : ( if_Code_integer @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7010_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > complex,H: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups6464643781859351333omplex
            @ ^ [J: nat] : ( if_complex @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_complex @ ( J = K2 ) @ one_one_complex @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups6464643781859351333omplex
            @ ^ [J: nat] : ( if_complex @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7011_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > real,H: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups129246275422532515t_real
            @ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_real @ ( J = K2 ) @ one_one_real @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups129246275422532515t_real
            @ ^ [J: nat] : ( if_real @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7012_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups705719431365010083at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_int @ ( J = K2 ) @ one_one_int @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups705719431365010083at_int
            @ ^ [J: nat] : ( if_int @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7013_prod_Ozero__middle,axiom,
    ! [P: nat,K2: nat,G2: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P )
     => ( ( ord_less_eq_nat @ K2 @ P )
       => ( ( groups708209901874060359at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( if_nat @ ( J = K2 ) @ one_one_nat @ ( H @ ( minus_minus_nat @ J @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P ) )
          = ( groups708209901874060359at_nat
            @ ^ [J: nat] : ( if_nat @ ( ord_less_nat @ J @ K2 ) @ ( G2 @ J ) @ ( H @ J ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% prod.zero_middle
thf(fact_7014_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J ) @ 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_7015_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J ) @ 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_7016_gbinomial__sum__up__index,axiom,
    ! [K2: nat,N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J ) @ 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_7017_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_7018_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_7019_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_7020_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_complex @ A @ K2 )
        = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7021_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_rat @ A @ K2 )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7022_gbinomial__absorption_H,axiom,
    ! [K2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ K2 )
     => ( ( gbinomial_real @ A @ K2 )
        = ( times_times_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( minus_minus_nat @ K2 @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_7023_sum__gp0,axiom,
    ! [X: complex,N: nat] :
      ( ( ( X = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X @ ( suc @ N ) ) ) @ ( minus_minus_complex @ one_one_complex @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7024_sum__gp0,axiom,
    ! [X: rat,N: nat] :
      ( ( ( X = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X @ ( suc @ N ) ) ) @ ( minus_minus_rat @ one_one_rat @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7025_sum__gp0,axiom,
    ! [X: real,N: nat] :
      ( ( ( X = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) )
      & ( ( X != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X ) @ ( set_ord_atMost_nat @ N ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( suc @ N ) ) ) @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% sum_gp0
thf(fact_7026_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X @ J ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7027_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X @ J ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7028_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X @ J ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7029_polyfun__diff__alt,axiom,
    ! [N: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X @ J ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ J ) ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_7030_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_7031_choose__linear__sum,axiom,
    ! [N: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I: nat] : ( times_times_nat @ I @ ( binomial @ N @ I ) )
        @ ( set_ord_atMost_nat @ N ) )
      = ( times_times_nat @ N @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_7032_card__lists__length__le,axiom,
    ! [A4: set_list_nat,N: nat] :
      ( ( finite8100373058378681591st_nat @ A4 )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs: list_list_nat] :
                ( ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A4 )
                & ( ord_less_eq_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_list_nat @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7033_card__lists__length__le,axiom,
    ! [A4: set_set_nat,N: nat] :
      ( ( finite1152437895449049373et_nat @ A4 )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs: list_set_nat] :
                ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 )
                & ( ord_less_eq_nat @ ( size_s3254054031482475050et_nat @ Xs ) @ N ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_set_nat @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7034_card__lists__length__le,axiom,
    ! [A4: set_complex,N: nat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs: list_complex] :
                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 )
                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs ) @ N ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_complex @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7035_card__lists__length__le,axiom,
    ! [A4: set_VEBT_VEBT,N: nat] :
      ( ( finite5795047828879050333T_VEBT @ A4 )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs: list_VEBT_VEBT] :
                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 )
                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ N ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite7802652506058667612T_VEBT @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7036_card__lists__length__le,axiom,
    ! [A4: set_o,N: nat] :
      ( ( finite_finite_o @ A4 )
     => ( ( finite_card_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 ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_o @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7037_card__lists__length__le,axiom,
    ! [A4: set_int,N: nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( finite_card_list_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 ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_int @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7038_card__lists__length__le,axiom,
    ! [A4: set_nat,N: nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( finite_card_list_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 ) ) ) )
        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( finite_card_nat @ A4 ) ) @ ( set_ord_atMost_nat @ N ) ) ) ) ).

% card_lists_length_le
thf(fact_7039_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: nat > complex,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z5: complex] :
          ( ( ord_less_eq_real @ M8 @ ( real_V1022390504157884413omplex @ Z5 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I: nat] : ( times_times_complex @ ( C2 @ I ) @ ( power_power_complex @ Z5 @ I ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7040_polyfun__extremal__lemma,axiom,
    ! [E: real,C2: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ E )
     => ? [M8: real] :
        ! [Z5: real] :
          ( ( ord_less_eq_real @ M8 @ ( real_V7735802525324610683m_real @ Z5 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( C2 @ I ) @ ( power_power_real @ Z5 @ I ) )
                @ ( set_ord_atMost_nat @ N ) ) )
            @ ( times_times_real @ E @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_7041_polyfun__diff,axiom,
    ! [N: nat,A: nat > complex,X: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I: nat] : ( times_times_complex @ ( A @ I ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
                @ ( power_power_complex @ X @ J ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7042_polyfun__diff,axiom,
    ! [N: nat,A: nat > rat,X: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
                @ ( power_power_rat @ X @ J ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7043_polyfun__diff,axiom,
    ! [N: nat,A: nat > int,X: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_int @ ( minus_minus_int @ X @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I: nat] : ( times_times_int @ ( A @ I ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
                @ ( power_power_int @ X @ J ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7044_polyfun__diff,axiom,
    ! [N: nat,A: nat > real,X: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ X @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ Y @ I ) )
            @ ( set_ord_atMost_nat @ N ) ) )
        = ( times_times_real @ ( minus_minus_real @ X @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( A @ I ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J ) @ N ) )
                @ ( power_power_real @ X @ J ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) ) ) ).

% polyfun_diff
thf(fact_7045_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_complex] :
      ( ( ord_less_nat @ K2 @ ( finite_card_complex @ A4 ) )
     => ( ( finite5120063068150530198omplex
          @ ( collect_list_complex
            @ ^ [Xs: list_complex] :
                ( ( ( size_s3451745648224563538omplex @ Xs )
                  = K2 )
                & ( distinct_complex @ Xs )
                & ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_complex @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_complex @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7046_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_list_nat] :
      ( ( ord_less_nat @ K2 @ ( finite_card_list_nat @ A4 ) )
     => ( ( finite7325466520557071688st_nat
          @ ( collec5989764272469232197st_nat
            @ ^ [Xs: list_list_nat] :
                ( ( ( size_s3023201423986296836st_nat @ Xs )
                  = K2 )
                & ( distinct_list_nat @ Xs )
                & ( ord_le6045566169113846134st_nat @ ( set_list_nat2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_list_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_list_nat @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7047_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_set_nat] :
      ( ( ord_less_nat @ K2 @ ( finite_card_set_nat @ A4 ) )
     => ( ( finite5631907774883551598et_nat
          @ ( collect_list_set_nat
            @ ^ [Xs: list_set_nat] :
                ( ( ( size_s3254054031482475050et_nat @ Xs )
                  = K2 )
                & ( distinct_set_nat @ Xs )
                & ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_set_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_set_nat @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7048_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_VEBT_VEBT] :
      ( ( ord_less_nat @ K2 @ ( finite7802652506058667612T_VEBT @ A4 ) )
     => ( ( finite5915292604075114978T_VEBT
          @ ( collec5608196760682091941T_VEBT
            @ ^ [Xs: list_VEBT_VEBT] :
                ( ( ( size_s6755466524823107622T_VEBT @ Xs )
                  = K2 )
                & ( distinct_VEBT_VEBT @ Xs )
                & ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite7802652506058667612T_VEBT @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite7802652506058667612T_VEBT @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7049_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_o] :
      ( ( ord_less_nat @ K2 @ ( finite_card_o @ A4 ) )
     => ( ( finite_card_list_o
          @ ( collect_list_o
            @ ^ [Xs: list_o] :
                ( ( ( size_size_list_o @ Xs )
                  = K2 )
                & ( distinct_o @ Xs )
                & ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_o @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_o @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7050_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_int] :
      ( ( ord_less_nat @ K2 @ ( finite_card_int @ A4 ) )
     => ( ( finite_card_list_int
          @ ( collect_list_int
            @ ^ [Xs: list_int] :
                ( ( ( size_size_list_int @ Xs )
                  = K2 )
                & ( distinct_int @ Xs )
                & ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_int @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_int @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7051_card__lists__distinct__length__eq_H,axiom,
    ! [K2: nat,A4: set_nat] :
      ( ( ord_less_nat @ K2 @ ( finite_card_nat @ A4 ) )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [Xs: list_nat] :
                ( ( ( size_size_list_nat @ Xs )
                  = K2 )
                & ( distinct_nat @ Xs )
                & ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 ) ) ) )
        = ( groups708209901874060359at_nat
          @ ^ [X4: nat] : X4
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( finite_card_nat @ A4 ) @ K2 ) @ one_one_nat ) @ ( finite_card_nat @ A4 ) ) ) ) ) ).

% card_lists_distinct_length_eq'
thf(fact_7052_Divides_Oadjust__div__eq,axiom,
    ! [Q2: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q2 @ R3 ) )
      = ( plus_plus_int @ Q2 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_7053_list__decode_Opinduct,axiom,
    ! [A0: nat,P2: nat > $o] :
      ( ( accp_nat @ nat_list_decode_rel @ A0 )
     => ( ( ( accp_nat @ nat_list_decode_rel @ zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
       => ( ! [N3: nat] :
              ( ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) )
             => ( ! [X5: nat,Y6: nat] :
                    ( ( ( product_Pair_nat_nat @ X5 @ Y6 )
                      = ( nat_prod_decode @ N3 ) )
                   => ( P2 @ Y6 ) )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ A0 ) ) ) ) ).

% list_decode.pinduct
thf(fact_7054_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] :
          ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
          @ ( divide1717551699836669952omplex
            @ ( set_fo1517530859248394432omplex
              @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A5 @ ( semiri8010041392384452111omplex @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7055_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A5: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A5 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7056_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A5: real,K3: nat] :
          ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
          @ ( divide_divide_real
            @ ( set_fo3111899725591712190t_real
              @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A5 @ ( semiri5074537144036343181t_real @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_7057_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_7058_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_7059_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_7060_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_7061_choose__alternating__sum,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_7062_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_7063_set__decode__0,axiom,
    ! [X: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X ) ) ) ).

% set_decode_0
thf(fact_7064_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_7065_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_7066_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_7067_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_7068_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_7069_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_7070_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_7071_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_7072_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_7073_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_7074_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N4: nat] : N4 ) ) ).

% of_nat_id
thf(fact_7075_compl__le__compl__iff,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ ( uminus5710092332889474511et_nat @ Y ) )
      = ( ord_less_eq_set_nat @ Y @ X ) ) ).

% compl_le_compl_iff
thf(fact_7076_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_7077_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_7078_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_7079_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_7080_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_7081_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_7082_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_7083_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_7084_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_7085_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_7086_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_7087_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_7088_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_7089_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_7090_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_7091_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_7092_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_7093_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_7094_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_7095_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_7096_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_7097_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_7098_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_7099_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_7100_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_7101_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_7102_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_7103_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_7104_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_7105_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_7106_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_7107_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_7108_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_7109_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_7110_neg__numeral__eq__iff,axiom,
    ! [M2: num,N: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( M2 = N ) ) ).

% neg_numeral_eq_iff
thf(fact_7111_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_7112_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7113_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7114_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7115_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7116_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_7117_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7118_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7119_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7120_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7121_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_7122_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7123_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7124_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7125_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7126_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_7127_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_7128_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_7129_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7130_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_7131_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_7132_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_7133_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_7134_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7135_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_7136_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_7137_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_7138_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_7139_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_7140_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_7141_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_7142_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_7143_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_7144_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7145_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_7146_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_7147_dvd__minus__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ X @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7148_dvd__minus__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ X @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7149_dvd__minus__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7150_dvd__minus__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7151_dvd__minus__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% dvd_minus_iff
thf(fact_7152_minus__dvd__iff,axiom,
    ! [X: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X ) @ Y )
      = ( dvd_dvd_int @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7153_minus__dvd__iff,axiom,
    ! [X: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X ) @ Y )
      = ( dvd_dvd_real @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7154_minus__dvd__iff,axiom,
    ! [X: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X ) @ Y )
      = ( dvd_dvd_complex @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7155_minus__dvd__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X ) @ Y )
      = ( dvd_dvd_Code_integer @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7156_minus__dvd__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X ) @ Y )
      = ( dvd_dvd_rat @ X @ Y ) ) ).

% minus_dvd_iff
thf(fact_7157_negative__eq__positive,axiom,
    ! [N: nat,M2: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
        = ( semiri1314217659103216013at_int @ M2 ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7158_signed__take__bit__of__0,axiom,
    ! [N: nat] :
      ( ( bit_ri631733984087533419it_int @ N @ zero_zero_int )
      = zero_zero_int ) ).

% signed_take_bit_of_0
thf(fact_7159_set__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_set_encode @ ( nat_set_decode @ N ) )
      = N ) ).

% set_decode_inverse
thf(fact_7160_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_7161_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_7162_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_7163_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_7164_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_7165_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_7166_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_7167_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_7168_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_7169_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_7170_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_7171_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_7172_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_7173_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_7174_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_7175_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_7176_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_7177_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_7178_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_7179_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_7180_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_7181_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_7182_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_7183_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_7184_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_7185_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_7186_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_7187_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_7188_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_7189_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_7190_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_7191_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_7192_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_7193_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_7194_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_7195_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_7196_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_7197_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_7198_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_7199_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_7200_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_7201_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_7202_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7203_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7204_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7205_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7206_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_7207_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_7208_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_7209_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_7210_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_7211_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_7212_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_7213_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_7214_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_7215_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_7216_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_7217_mult__minus1__right,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ Z3 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z3 ) ) ).

% mult_minus1_right
thf(fact_7218_mult__minus1__right,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ Z3 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z3 ) ) ).

% mult_minus1_right
thf(fact_7219_mult__minus1__right,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ Z3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z3 ) ) ).

% mult_minus1_right
thf(fact_7220_mult__minus1__right,axiom,
    ! [Z3: code_integer] :
      ( ( times_3573771949741848930nteger @ Z3 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z3 ) ) ).

% mult_minus1_right
thf(fact_7221_mult__minus1__right,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ Z3 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z3 ) ) ).

% mult_minus1_right
thf(fact_7222_mult__minus1,axiom,
    ! [Z3: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z3 )
      = ( uminus_uminus_int @ Z3 ) ) ).

% mult_minus1
thf(fact_7223_mult__minus1,axiom,
    ! [Z3: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z3 )
      = ( uminus_uminus_real @ Z3 ) ) ).

% mult_minus1
thf(fact_7224_mult__minus1,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z3 )
      = ( uminus1482373934393186551omplex @ Z3 ) ) ).

% mult_minus1
thf(fact_7225_mult__minus1,axiom,
    ! [Z3: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z3 )
      = ( uminus1351360451143612070nteger @ Z3 ) ) ).

% mult_minus1
thf(fact_7226_mult__minus1,axiom,
    ! [Z3: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z3 )
      = ( uminus_uminus_rat @ Z3 ) ) ).

% mult_minus1
thf(fact_7227_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_7228_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_7229_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_7230_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_7231_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_7232_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_7233_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_7234_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_7235_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_7236_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_7237_fact__0,axiom,
    ( ( semiri3624122377584611663nteger @ zero_zero_nat )
    = one_one_Code_integer ) ).

% fact_0
thf(fact_7238_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_7239_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_7240_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_7241_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_7242_negative__zless,axiom,
    ! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).

% negative_zless
thf(fact_7243_signed__take__bit__Suc__1,axiom,
    ! [N: nat] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N ) @ one_one_Code_integer )
      = one_one_Code_integer ) ).

% signed_take_bit_Suc_1
thf(fact_7244_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_7245_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_7246_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_7247_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_7248_dbl__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_simps(1)
thf(fact_7249_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_7250_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_7251_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_7252_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_7253_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_7254_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_7255_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_7256_set__encode__inverse,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
        = A4 ) ) ).

% set_encode_inverse
thf(fact_7257_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_7258_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_7259_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_7260_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_7261_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_7262_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_7263_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_7264_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_7265_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_7266_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_7267_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% diff_numeral_special(12)
thf(fact_7268_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
    = zero_zero_real ) ).

% diff_numeral_special(12)
thf(fact_7269_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = zero_zero_complex ) ).

% diff_numeral_special(12)
thf(fact_7270_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_7271_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_7272_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_7273_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_7274_neg__one__eq__numeral__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) )
      = ( N = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_7275_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_7276_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_7277_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_7278_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_7279_numeral__eq__neg__one__iff,axiom,
    ! [N: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_7280_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_7281_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_7282_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_7283_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_7284_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_7285_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_7286_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_7287_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_7288_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_7289_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_7290_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_7291_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_7292_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_7293_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_7294_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_7295_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_7296_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_7297_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_7298_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_7299_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_7300_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_7301_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_7302_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_7303_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_7304_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_7305_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_7306_fact__Suc__0,axiom,
    ( ( semiri3624122377584611663nteger @ ( suc @ zero_zero_nat ) )
    = one_one_Code_integer ) ).

% fact_Suc_0
thf(fact_7307_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_7308_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_7309_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_7310_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_7311_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ).

% fact_Suc
thf(fact_7312_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ).

% fact_Suc
thf(fact_7313_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N ) ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ).

% fact_Suc
thf(fact_7314_fact__Suc,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ).

% fact_Suc
thf(fact_7315_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_7316_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_7317_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_7318_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_7319_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_7320_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_7321_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_7322_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_7323_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_7324_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_7325_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_7326_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_7327_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_7328_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_7329_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_7330_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_7331_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_7332_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_7333_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_7334_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_7335_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_7336_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_7337_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_7338_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_7339_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_7340_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7341_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7342_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7343_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7344_semiring__norm_I172_J,axiom,
    ! [V2: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W2 ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_7345_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V2 ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7346_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V2 ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7347_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V2 ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7348_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V2 ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7349_semiring__norm_I171_J,axiom,
    ! [V2: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V2 ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_7350_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V2 ) ) @ ( times_times_int @ ( numeral_numeral_int @ W2 ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7351_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V2 ) ) @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7352_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V2 ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7353_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V2 ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7354_semiring__norm_I170_J,axiom,
    ! [V2: num,W2: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V2 ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V2 @ W2 ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_7355_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_7356_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_7357_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_7358_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_7359_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_7360_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_7361_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_7362_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_7363_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_7364_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_7365_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_7366_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_7367_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_7368_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_7369_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_7370_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_7371_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_7372_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7373_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_7374_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7375_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_7376_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7377_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7378_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_7379_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7380_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7381_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_7382_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_7383_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_7384_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_7385_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_7386_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7387_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_7388_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7389_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_7390_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_7391_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_7392_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7393_dbl__dec__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_7394_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_7395_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_7396_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_7397_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K2 ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7398_dbl__inc__simps_I1_J,axiom,
    ! [K2: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K2 ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K2 ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_7399_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_7400_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_7401_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_7402_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_7403_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_7404_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_7405_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_7406_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_7407_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_7408_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_7409_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_7410_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_7411_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_7412_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_7413_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_7414_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_7415_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_7416_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_7417_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_7418_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_7419_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_7420_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_7421_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_7422_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_7423_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_7424_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_7425_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_7426_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_7427_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_7428_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_7429_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_7430_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_7431_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_7432_set__decode__Suc,axiom,
    ! [N: nat,X: nat] :
      ( ( member_nat @ ( suc @ N ) @ ( nat_set_decode @ X ) )
      = ( member_nat @ N @ ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_7433_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_7434_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_7435_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7436_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_7437_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_7438_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_7439_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_7440_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_7441_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_7442_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_7443_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_7444_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_7445_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_7446_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_7447_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_7448_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_7449_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_7450_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_7451_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_7452_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_7453_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_7454_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_7455_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_7456_fact__ge__self,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_self
thf(fact_7457_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri5044797733671781792omplex @ N )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_7458_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri773545260158071498ct_rat @ N )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_7459_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1406184849735516958ct_int @ N )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_7460_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri1408675320244567234ct_nat @ N )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_7461_fact__nonzero,axiom,
    ! [N: nat] :
      ( ( semiri2265585572941072030t_real @ N )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_7462_compl__le__swap2,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ X )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ X ) @ Y ) ) ).

% compl_le_swap2
thf(fact_7463_compl__le__swap1,axiom,
    ! [Y: set_nat,X: set_nat] :
      ( ( ord_less_eq_set_nat @ Y @ ( uminus5710092332889474511et_nat @ X ) )
     => ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% compl_le_swap1
thf(fact_7464_compl__mono,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ord_less_eq_set_nat @ X @ Y )
     => ( ord_less_eq_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ ( uminus5710092332889474511et_nat @ X ) ) ) ).

% compl_mono
thf(fact_7465_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_7466_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_7467_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_7468_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_7469_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_7470_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_7471_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_7472_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_7473_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_7474_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_7475_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_7476_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_7477_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_7478_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_7479_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_7480_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_7481_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_7482_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_7483_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_7484_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_7485_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_7486_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_7487_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M2 ) )
     != ( numera6690914467698888265omplex @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_7488_neg__numeral__neq__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) )
     != ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_neq_numeral
thf(fact_7489_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_7490_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_7491_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_7492_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6690914467698888265omplex @ M2 )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7493_numeral__neq__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( numera6620942414471956472nteger @ M2 )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_7494_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_7495_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_7496_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_7497_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_7498_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_7499_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_7500_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_7501_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_7502_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_7503_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_7504_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_7505_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_7506_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_7507_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_7508_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_7509_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_7510_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_7511_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_7512_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_7513_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_7514_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_7515_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_7516_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_7517_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_7518_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_7519_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_7520_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_7521_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_7522_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_7523_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_7524_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_7525_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_7526_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_7527_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_7528_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_7529_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_7530_signed__take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N @ K2 ) @ ( bit_ri631733984087533419it_int @ N @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N @ ( times_times_int @ K2 @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_7531_fact__less__mono__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono_nat
thf(fact_7532_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_zero
thf(fact_7533_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_zero
thf(fact_7534_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_zero
thf(fact_7535_fact__ge__zero,axiom,
    ! [N: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_zero
thf(fact_7536_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_7537_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_7538_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_7539_fact__not__neg,axiom,
    ! [N: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_7540_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_gt_zero
thf(fact_7541_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_gt_zero
thf(fact_7542_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_gt_zero
thf(fact_7543_fact__gt__zero,axiom,
    ! [N: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_gt_zero
thf(fact_7544_finite__set__decode,axiom,
    ! [N: nat] : ( finite_finite_nat @ ( nat_set_decode @ N ) ) ).

% finite_set_decode
thf(fact_7545_fact__ge__1,axiom,
    ! [N: nat] : ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( semiri3624122377584611663nteger @ N ) ) ).

% fact_ge_1
thf(fact_7546_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N ) ) ).

% fact_ge_1
thf(fact_7547_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N ) ) ).

% fact_ge_1
thf(fact_7548_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N ) ) ).

% fact_ge_1
thf(fact_7549_fact__ge__1,axiom,
    ! [N: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N ) ) ).

% fact_ge_1
thf(fact_7550_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_7551_neg__numeral__le__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_le_numeral
thf(fact_7552_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_7553_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_7554_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_7555_not__numeral__le__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_7556_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_7557_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_7558_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7559_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7560_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7561_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7562_zero__neq__neg__numeral,axiom,
    ! [N: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% zero_neq_neg_numeral
thf(fact_7563_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_7564_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_7565_not__numeral__less__neg__numeral,axiom,
    ! [M2: num,N: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_7566_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_7567_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_7568_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_7569_neg__numeral__less__numeral,axiom,
    ! [M2: num,N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( numera6620942414471956472nteger @ N ) ) ).

% neg_numeral_less_numeral
thf(fact_7570_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_7571_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_7572_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_7573_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_7574_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_7575_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_7576_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_7577_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_7578_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_7579_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_7580_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_7581_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_7582_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_7583_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_7584_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_7585_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_7586_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_7587_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_7588_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_7589_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_7590_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_7591_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_7592_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_7593_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_7594_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_7595_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_7596_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_7597_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_7598_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_7599_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_7600_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_7601_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_7602_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_7603_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_7604_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_7605_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_7606_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_7607_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_7608_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_7609_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(4)
thf(fact_7610_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(4)
thf(fact_7611_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_7612_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_7613_less__minus__one__simps_I2_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).

% less_minus_one_simps(2)
thf(fact_7614_less__minus__one__simps_I2_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).

% less_minus_one_simps(2)
thf(fact_7615_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_7616_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_7617_numeral__times__minus__swap,axiom,
    ! [W2: num,X: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W2 ) @ ( uminus_uminus_int @ X ) )
      = ( times_times_int @ X @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_7618_numeral__times__minus__swap,axiom,
    ! [W2: num,X: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ ( uminus_uminus_real @ X ) )
      = ( times_times_real @ X @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_7619_numeral__times__minus__swap,axiom,
    ! [W2: num,X: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ ( uminus1482373934393186551omplex @ X ) )
      = ( times_times_complex @ X @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_7620_numeral__times__minus__swap,axiom,
    ! [W2: num,X: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W2 ) @ ( uminus1351360451143612070nteger @ X ) )
      = ( times_3573771949741848930nteger @ X @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_7621_numeral__times__minus__swap,axiom,
    ! [W2: num,X: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ ( uminus_uminus_rat @ X ) )
      = ( times_times_rat @ X @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_7622_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7623_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7624_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_7625_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7626_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7627_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_7628_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_7629_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_7630_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_7631_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_7632_one__neq__neg__numeral,axiom,
    ! [N: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N ) ) ) ).

% one_neq_neg_numeral
thf(fact_7633_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_int @ N )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_7634_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_real @ N )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_7635_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6690914467698888265omplex @ N )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_7636_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ N )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_7637_numeral__neq__neg__one,axiom,
    ! [N: num] :
      ( ( numeral_numeral_rat @ N )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_7638_square__eq__1__iff,axiom,
    ! [X: int] :
      ( ( ( times_times_int @ X @ X )
        = one_one_int )
      = ( ( X = one_one_int )
        | ( X
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_7639_square__eq__1__iff,axiom,
    ! [X: real] :
      ( ( ( times_times_real @ X @ X )
        = one_one_real )
      = ( ( X = one_one_real )
        | ( X
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_7640_square__eq__1__iff,axiom,
    ! [X: complex] :
      ( ( ( times_times_complex @ X @ X )
        = one_one_complex )
      = ( ( X = one_one_complex )
        | ( X
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_7641_square__eq__1__iff,axiom,
    ! [X: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X @ X )
        = one_one_Code_integer )
      = ( ( X = one_one_Code_integer )
        | ( X
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_7642_square__eq__1__iff,axiom,
    ! [X: rat] :
      ( ( ( times_times_rat @ X @ X )
        = one_one_rat )
      = ( ( X = one_one_rat )
        | ( X
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_7643_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_7644_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_7645_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_7646_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_7647_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7648_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7649_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7650_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7651_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_7652_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A5: int,B4: int] : ( plus_plus_int @ A5 @ ( uminus_uminus_int @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7653_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A5: real,B4: real] : ( plus_plus_real @ A5 @ ( uminus_uminus_real @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7654_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A5: complex,B4: complex] : ( plus_plus_complex @ A5 @ ( uminus1482373934393186551omplex @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7655_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A5: code_integer,B4: code_integer] : ( plus_p5714425477246183910nteger @ A5 @ ( uminus1351360451143612070nteger @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7656_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A5: rat,B4: rat] : ( plus_plus_rat @ A5 @ ( uminus_uminus_rat @ B4 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_7657_group__cancel_Osub2,axiom,
    ! [B5: int,K2: int,B: int,A: int] :
      ( ( B5
        = ( plus_plus_int @ K2 @ B ) )
     => ( ( minus_minus_int @ A @ B5 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K2 ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_7658_group__cancel_Osub2,axiom,
    ! [B5: real,K2: real,B: real,A: real] :
      ( ( B5
        = ( plus_plus_real @ K2 @ B ) )
     => ( ( minus_minus_real @ A @ B5 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K2 ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_7659_group__cancel_Osub2,axiom,
    ! [B5: complex,K2: complex,B: complex,A: complex] :
      ( ( B5
        = ( plus_plus_complex @ K2 @ B ) )
     => ( ( minus_minus_complex @ A @ B5 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K2 ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_7660_group__cancel_Osub2,axiom,
    ! [B5: code_integer,K2: code_integer,B: code_integer,A: code_integer] :
      ( ( B5
        = ( plus_p5714425477246183910nteger @ K2 @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B5 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K2 ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_7661_group__cancel_Osub2,axiom,
    ! [B5: rat,K2: rat,B: rat,A: rat] :
      ( ( B5
        = ( plus_plus_rat @ K2 @ B ) )
     => ( ( minus_minus_rat @ A @ B5 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K2 ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_7662_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_7663_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_7664_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_7665_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_7666_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_7667_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_7668_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_7669_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_7670_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_7671_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_7672_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_7673_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_7674_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_7675_real__minus__mult__self__le,axiom,
    ! [U: real,X: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X @ X ) ) ).

% real_minus_mult_self_le
thf(fact_7676_int__of__nat__induct,axiom,
    ! [P2: int > $o,Z3: int] :
      ( ! [N3: nat] : ( P2 @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ! [N3: nat] : ( P2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
       => ( P2 @ Z3 ) ) ) ).

% int_of_nat_induct
thf(fact_7677_int__cases,axiom,
    ! [Z3: int] :
      ( ! [N3: nat] :
          ( Z3
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z3
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% int_cases
thf(fact_7678_zmult__eq__1__iff,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
      = ( ( ( M2 = one_one_int )
          & ( N = one_one_int ) )
        | ( ( M2
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_7679_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M2: int,N: int] :
      ( ( ( times_times_int @ M2 @ N )
        = one_one_int )
     => ( ( M2 = one_one_int )
        | ( M2
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_7680_zmod__zminus1__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K2 @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_7681_zmod__zminus2__not__zero,axiom,
    ! [K2: int,L: int] :
      ( ( ( modulo_modulo_int @ K2 @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K2 @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_7682_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_7683_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_7684_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_7685_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_7686_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_7687_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_7688_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_7689_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_7690_not__zero__le__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_7691_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_7692_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_7693_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_7694_neg__numeral__le__zero,axiom,
    ! [N: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_7695_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_7696_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_7697_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_7698_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_7699_not__zero__less__neg__numeral,axiom,
    ! [N: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_7700_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_7701_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_7702_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_7703_neg__numeral__less__zero,axiom,
    ! [N: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_7704_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_7705_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_7706_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_7707_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_7708_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_7709_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_7710_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_7711_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_7712_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_7713_less__minus__one__simps_I1_J,axiom,
    ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).

% less_minus_one_simps(1)
thf(fact_7714_less__minus__one__simps_I1_J,axiom,
    ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).

% less_minus_one_simps(1)
thf(fact_7715_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_7716_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_7717_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).

% less_minus_one_simps(3)
thf(fact_7718_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).

% less_minus_one_simps(3)
thf(fact_7719_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_7720_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_7721_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_7722_not__one__le__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_le_neg_numeral
thf(fact_7723_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_7724_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_7725_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_7726_not__numeral__le__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_7727_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_7728_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_7729_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_7730_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_7731_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_7732_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_7733_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_7734_neg__one__le__numeral,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_le_numeral
thf(fact_7735_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_7736_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_7737_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_7738_neg__numeral__le__one,axiom,
    ! [M2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_7739_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_7740_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_7741_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A5 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7742_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A5: 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 @ A5 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7743_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A5: 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 @ A5 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_7744_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_7745_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_7746_neg__numeral__less__one,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_7747_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_7748_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_7749_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_7750_neg__one__less__numeral,axiom,
    ! [M2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M2 ) ) ).

% neg_one_less_numeral
thf(fact_7751_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_7752_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_7753_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_7754_not__numeral__less__neg__one,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_7755_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_7756_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_7757_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_7758_not__one__less__neg__numeral,axiom,
    ! [M2: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M2 ) ) ) ).

% not_one_less_neg_numeral
thf(fact_7759_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_7760_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_7761_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_7762_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_7763_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_7764_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ A @ C2 )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7765_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C2: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ A @ C2 )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7766_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ A @ C2 )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_7767_minus__divide__eq__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) )
        = A )
      = ( ( ( C2 != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7768_minus__divide__eq__eq,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C2 ) )
        = A )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7769_minus__divide__eq__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) )
        = A )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_7770_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C2: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C2 )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C2 @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7771_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C2: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C2 )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C2 @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7772_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C2: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C2 )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C2 @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_7773_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C2
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C2 @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7774_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C2: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C2
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C2 @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7775_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C2
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C2 @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_7776_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_7777_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_7778_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_7779_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_7780_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_7781_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_7782_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_7783_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_7784_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_7785_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_7786_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7787_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7788_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_7789_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_7790_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_7791_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_7792_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_7793_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_7794_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7795_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M2 ) @ ( semiri1406184849735516958ct_int @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7796_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7797_fact__less__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ M2 @ N )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M2 ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ).

% fact_less_mono
thf(fact_7798_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_7799_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_7800_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_7801_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_7802_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_7803_inf__shunt,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat] :
      ( ( ( inf_in2572325071724192079at_nat @ X @ Y )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_7804_inf__shunt,axiom,
    ! [X: set_int,Y: set_int] :
      ( ( ( inf_inf_set_int @ X @ Y )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% inf_shunt
thf(fact_7805_inf__shunt,axiom,
    ! [X: set_real,Y: set_real] :
      ( ( ( inf_inf_set_real @ X @ Y )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X @ ( uminus612125837232591019t_real @ Y ) ) ) ).

% inf_shunt
thf(fact_7806_inf__shunt,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( ( inf_inf_set_nat @ X @ Y )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) ) ).

% inf_shunt
thf(fact_7807_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_7808_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_7809_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_7810_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_7811_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_7812_fact__mod,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( modulo8411746178871703098atural @ ( semiri2447717529341329178atural @ N ) @ ( semiri2447717529341329178atural @ M2 ) )
        = zero_z2226904508553997617atural ) ) ).

% fact_mod
thf(fact_7813_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_7814_sup__neg__inf,axiom,
    ! [P: set_Pr1261947904930325089at_nat,Q2: set_Pr1261947904930325089at_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ P @ ( sup_su6327502436637775413at_nat @ Q2 @ R3 ) )
      = ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ P @ ( uminus6524753893492686040at_nat @ Q2 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_7815_sup__neg__inf,axiom,
    ! [P: set_nat,Q2: set_nat,R3: set_nat] :
      ( ( ord_less_eq_set_nat @ P @ ( sup_sup_set_nat @ Q2 @ R3 ) )
      = ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ P @ ( uminus5710092332889474511et_nat @ Q2 ) ) @ R3 ) ) ).

% sup_neg_inf
thf(fact_7816_shunt2,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ ( uminus6524753893492686040at_nat @ Y ) ) @ Z3 )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_7817_shunt2,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ ( uminus5710092332889474511et_nat @ Y ) ) @ Z3 )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z3 ) ) ) ).

% shunt2
thf(fact_7818_shunt1,axiom,
    ! [X: set_Pr1261947904930325089at_nat,Y: set_Pr1261947904930325089at_nat,Z3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( inf_in2572325071724192079at_nat @ X @ Y ) @ Z3 )
      = ( ord_le3146513528884898305at_nat @ X @ ( sup_su6327502436637775413at_nat @ ( uminus6524753893492686040at_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_7819_shunt1,axiom,
    ! [X: set_nat,Y: set_nat,Z3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z3 )
      = ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ ( uminus5710092332889474511et_nat @ Y ) @ Z3 ) ) ) ).

% shunt1
thf(fact_7820_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_7821_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_7822_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_7823_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_7824_int__cases4,axiom,
    ! [M2: int] :
      ( ! [N3: nat] :
          ( M2
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M2
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_7825_int__zle__neg,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
      = ( ( N = zero_zero_nat )
        & ( M2 = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_7826_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_7827_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_7828_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N: nat] :
      ( ( ord_less_eq_nat @ R3 @ N )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ R3 ) ) ) @ ( power_power_nat @ N @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_7829_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_7830_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_7831_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_7832_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_7833_minus__divide__less__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_7834_minus__divide__less__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_7835_neg__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_7836_neg__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_7837_neg__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_7838_neg__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_7839_pos__less__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_7840_pos__less__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_7841_pos__minus__divide__less__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_7842_pos__minus__divide__less__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_7843_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7844_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C2: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7845_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 )
            = B ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_7846_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C2 )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C2 != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7847_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C2: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C2 )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C2 != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7848_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C2 )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C2 != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
        & ( ( C2 = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_7849_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7850_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7851_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = B ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_7852_minus__divide__add__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z3 ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7853_minus__divide__add__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z3 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7854_minus__divide__add__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z3 ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_7855_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_7856_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z3 ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7857_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z3 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7858_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z3 ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_7859_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z3: real,A: real,B: real] :
      ( ( ( Z3 = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z3 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z3 ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7860_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z3: complex,A: complex,B: complex] :
      ( ( ( Z3 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z3 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z3 ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7861_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z3: rat,A: rat,B: rat] :
      ( ( ( Z3 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z3 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z3 ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z3 ) ) @ Z3 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_7862_minus__divide__diff__eq__iff,axiom,
    ! [Z3: real,X: real,Y: real] :
      ( ( Z3 != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X @ Z3 ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7863_minus__divide__diff__eq__iff,axiom,
    ! [Z3: complex,X: complex,Y: complex] :
      ( ( Z3 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X @ Z3 ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X ) @ ( times_times_complex @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7864_minus__divide__diff__eq__iff,axiom,
    ! [Z3: rat,X: rat,Y: rat] :
      ( ( Z3 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X @ Z3 ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X ) @ ( times_times_rat @ Y @ Z3 ) ) @ Z3 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_7865_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_7866_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_7867_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_7868_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_7869_int__cases3,axiom,
    ! [K2: int] :
      ( ( K2 != zero_zero_int )
     => ( ! [N3: nat] :
            ( ( K2
              = ( semiri1314217659103216013at_int @ N3 ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
       => ~ ! [N3: nat] :
              ( ( K2
                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).

% int_cases3
thf(fact_7870_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
        = zero_zero_complex ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7871_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7872_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7873_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
        = zero_zero_real ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7874_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N: nat,K2: nat] :
      ( ( ord_less_nat @ N @ K2 )
     => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
        = zero_zero_int ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_7875_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N ) ) @ K2 )
        = zero_zero_complex )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7876_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N ) ) @ K2 )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7877_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N ) ) @ K2 )
        = zero_zero_rat )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7878_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ K2 )
        = zero_zero_real )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7879_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N: nat,K2: nat] :
      ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ K2 )
        = zero_zero_int )
      = ( ord_less_nat @ N @ K2 ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_7880_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7881_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7882_pochhammer__eq__0__iff,axiom,
    ! [A: real,N: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_7883_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_7884_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_7885_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_7886_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_7887_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_7888_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_7889_negD,axiom,
    ! [X: int] :
      ( ( ord_less_int @ X @ zero_zero_int )
     => ? [N3: nat] :
          ( X
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_7890_negative__zless__0,axiom,
    ! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7891_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_7892_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: real > $o,H: real > real,G2: real > real] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups1681761925125756287l_real
          @ ^ [X4: real] : ( if_real @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_real @ ( groups1681761925125756287l_real @ H @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) @ ( groups1681761925125756287l_real @ G2 @ ( inf_inf_set_real @ A4 @ ( uminus612125837232591019t_real @ ( collect_real @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7893_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > real,G2: int > real] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups2316167850115554303t_real
          @ ^ [X4: int] : ( if_real @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_real @ ( groups2316167850115554303t_real @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups2316167850115554303t_real @ G2 @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7894_prod_OIf__cases,axiom,
    ! [A4: set_complex,P2: complex > $o,H: complex > real,G2: complex > real] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups766887009212190081x_real
          @ ^ [X4: complex] : ( if_real @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_real @ ( groups766887009212190081x_real @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) @ ( groups766887009212190081x_real @ G2 @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7895_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > real,G2: nat > real] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups129246275422532515t_real
          @ ^ [X4: nat] : ( if_real @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_real @ ( groups129246275422532515t_real @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups129246275422532515t_real @ G2 @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7896_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: real > $o,H: real > rat,G2: real > rat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4061424788464935467al_rat
          @ ^ [X4: real] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups4061424788464935467al_rat @ H @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) @ ( groups4061424788464935467al_rat @ G2 @ ( inf_inf_set_real @ A4 @ ( uminus612125837232591019t_real @ ( collect_real @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7897_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > rat,G2: int > rat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1072433553688619179nt_rat
          @ ^ [X4: int] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups1072433553688619179nt_rat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups1072433553688619179nt_rat @ G2 @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7898_prod_OIf__cases,axiom,
    ! [A4: set_complex,P2: complex > $o,H: complex > rat,G2: complex > rat] :
      ( ( finite3207457112153483333omplex @ A4 )
     => ( ( groups225925009352817453ex_rat
          @ ^ [X4: complex] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups225925009352817453ex_rat @ H @ ( inf_inf_set_complex @ A4 @ ( collect_complex @ P2 ) ) ) @ ( groups225925009352817453ex_rat @ G2 @ ( inf_inf_set_complex @ A4 @ ( uminus8566677241136511917omplex @ ( collect_complex @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7899_prod_OIf__cases,axiom,
    ! [A4: set_nat,P2: nat > $o,H: nat > rat,G2: nat > rat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( groups73079841787564623at_rat
          @ ^ [X4: nat] : ( if_rat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_rat @ ( groups73079841787564623at_rat @ H @ ( inf_inf_set_nat @ A4 @ ( collect_nat @ P2 ) ) ) @ ( groups73079841787564623at_rat @ G2 @ ( inf_inf_set_nat @ A4 @ ( uminus5710092332889474511et_nat @ ( collect_nat @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7900_prod_OIf__cases,axiom,
    ! [A4: set_real,P2: real > $o,H: real > nat,G2: real > nat] :
      ( ( finite_finite_real @ A4 )
     => ( ( groups4696554848551431203al_nat
          @ ^ [X4: real] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_nat @ ( groups4696554848551431203al_nat @ H @ ( inf_inf_set_real @ A4 @ ( collect_real @ P2 ) ) ) @ ( groups4696554848551431203al_nat @ G2 @ ( inf_inf_set_real @ A4 @ ( uminus612125837232591019t_real @ ( collect_real @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7901_prod_OIf__cases,axiom,
    ! [A4: set_int,P2: int > $o,H: int > nat,G2: int > nat] :
      ( ( finite_finite_int @ A4 )
     => ( ( groups1707563613775114915nt_nat
          @ ^ [X4: int] : ( if_nat @ ( P2 @ X4 ) @ ( H @ X4 ) @ ( G2 @ X4 ) )
          @ A4 )
        = ( times_times_nat @ ( groups1707563613775114915nt_nat @ H @ ( inf_inf_set_int @ A4 @ ( collect_int @ P2 ) ) ) @ ( groups1707563613775114915nt_nat @ G2 @ ( inf_inf_set_int @ A4 @ ( uminus1532241313380277803et_int @ ( collect_int @ P2 ) ) ) ) ) ) ) ).

% prod.If_cases
thf(fact_7902_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_7903_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_7904_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_7905_minus__divide__le__eq,axiom,
    ! [B: real,C2: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_7906_minus__divide__le__eq,axiom,
    ! [B: rat,C2: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_7907_neg__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_7908_neg__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_7909_neg__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ C2 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_7910_neg__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C2 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_7911_pos__le__minus__divide__eq,axiom,
    ! [C2: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C2 ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_7912_pos__le__minus__divide__eq,axiom,
    ! [C2: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C2 ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_7913_pos__minus__divide__le__eq,axiom,
    ! [C2: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C2 )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C2 ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_7914_pos__minus__divide__le__eq,axiom,
    ! [C2: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C2 )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C2 ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C2 ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_7915_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_7916_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_7917_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_7918_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_7919_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_7920_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_7921_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_7922_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_7923_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_7924_neg__int__cases,axiom,
    ! [K2: int] :
      ( ( ord_less_int @ K2 @ zero_zero_int )
     => ~ ! [N3: nat] :
            ( ( K2
              = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
           => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).

% neg_int_cases
thf(fact_7925_minus__mod__int__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K2 ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K2 @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_7926_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_7927_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_7928_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_7929_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_7930_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N4: nat,A5: code_integer] : ( if_Code_integer @ ( N4 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7931_signed__take__bit__rec,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,A5: int] : ( if_int @ ( N4 = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N4 @ one_one_nat ) @ ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_7932_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_7933_zminus1__lemma,axiom,
    ! [A: int,B: int,Q2: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q2 @ R3 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q2 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q2 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_7934_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C2: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C2 ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_7935_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C2: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C2 ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_7936_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C2: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C2 )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C2 )
         => ( ( ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_real @ C2 @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_7937_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C2: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C2 )
         => ( ( ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C2 ) ) )
            & ( ~ ( ord_less_rat @ C2 @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_7938_square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ( ord_less_eq_real @ X @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_7939_square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X )
     => ( ( ord_le3102999989581377725nteger @ X @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_7940_square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X )
     => ( ( ord_less_eq_rat @ X @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_7941_square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X )
     => ( ( ord_less_eq_int @ X @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_7942_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_7943_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_7944_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_7945_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_7946_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_7947_fact__num__eq__if,axiom,
    ( semiri3624122377584611663nteger
    = ( ^ [M6: nat] : ( if_Code_integer @ ( M6 = zero_zero_nat ) @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M6 ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7948_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7949_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7950_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7951_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7952_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_7953_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N4: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7954_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N4: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7955_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N4: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_7956_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri773545260158071498ct_rat @ N )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7957_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1406184849735516958ct_int @ N )
        = ( times_times_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7958_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri1408675320244567234ct_nat @ N )
        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7959_fact__reduce,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( semiri2265585572941072030t_real @ N )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_7960_pochhammer__absorb__comp,axiom,
    ! [R3: complex,K2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R3 @ ( semiri8010041392384452111omplex @ K2 ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R3 ) @ K2 ) )
      = ( times_times_complex @ R3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R3 ) @ one_one_complex ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7961_pochhammer__absorb__comp,axiom,
    ! [R3: code_integer,K2: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R3 @ ( semiri4939895301339042750nteger @ K2 ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R3 ) @ K2 ) )
      = ( times_3573771949741848930nteger @ R3 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7962_pochhammer__absorb__comp,axiom,
    ! [R3: rat,K2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K2 ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K2 ) )
      = ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7963_pochhammer__absorb__comp,axiom,
    ! [R3: real,K2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ K2 ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R3 ) @ K2 ) )
      = ( times_times_real @ R3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R3 ) @ one_one_real ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7964_pochhammer__absorb__comp,axiom,
    ! [R3: int,K2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K2 ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K2 ) )
      = ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K2 ) ) ) ).

% pochhammer_absorb_comp
thf(fact_7965_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_7966_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_7967_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_7968_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A5: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A5 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7969_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A5: 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 ) @ A5 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7970_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A5: 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 ) @ A5 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_7971_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_7972_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_7973_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_7974_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_7975_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_7976_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_7977_Bernoulli__inequality,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ X ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X ) @ N ) ) ) ).

% Bernoulli_inequality
thf(fact_7978_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_7979_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_7980_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_7981_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_7982_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_7983_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_7984_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_7985_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_7986_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_7987_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_7988_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_7989_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_7990_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_7991_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_7992_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_7993_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_7994_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_7995_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_7996_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_7997_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_7998_int__bit__induct,axiom,
    ! [P2: int > $o,K2: int] :
      ( ( P2 @ zero_zero_int )
     => ( ( P2 @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K: int] :
              ( ( P2 @ K )
             => ( ( K != zero_zero_int )
               => ( P2 @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K: int] :
                ( ( P2 @ K )
               => ( ( K
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P2 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P2 @ K2 ) ) ) ) ) ).

% int_bit_induct
thf(fact_7999_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_8000_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_8001_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_8002_gbinomial__Suc,axiom,
    ! [A: complex,K2: nat] :
      ( ( gbinomial_complex @ A @ ( suc @ K2 ) )
      = ( divide1717551699836669952omplex
        @ ( groups6464643781859351333omplex
          @ ^ [I: nat] : ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri5044797733671781792omplex @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8003_gbinomial__Suc,axiom,
    ! [A: rat,K2: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K2 ) )
      = ( divide_divide_rat
        @ ( groups73079841787564623at_rat
          @ ^ [I: nat] : ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri773545260158071498ct_rat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8004_gbinomial__Suc,axiom,
    ! [A: real,K2: nat] :
      ( ( gbinomial_real @ A @ ( suc @ K2 ) )
      = ( divide_divide_real
        @ ( groups129246275422532515t_real
          @ ^ [I: nat] : ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri2265585572941072030t_real @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8005_gbinomial__Suc,axiom,
    ! [A: int,K2: nat] :
      ( ( gbinomial_int @ A @ ( suc @ K2 ) )
      = ( divide_divide_int
        @ ( groups705719431365010083at_int
          @ ^ [I: nat] : ( minus_minus_int @ A @ ( semiri1314217659103216013at_int @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1406184849735516958ct_int @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8006_gbinomial__Suc,axiom,
    ! [A: nat,K2: nat] :
      ( ( gbinomial_nat @ A @ ( suc @ K2 ) )
      = ( divide_divide_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I: nat] : ( minus_minus_nat @ A @ ( semiri1316708129612266289at_nat @ I ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K2 ) )
        @ ( semiri1408675320244567234ct_nat @ ( suc @ K2 ) ) ) ) ).

% gbinomial_Suc
thf(fact_8007_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: complex,X: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M2 ) @ A ) @ K3 ) @ ( power_power_complex @ X @ 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 @ X ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8008_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: rat,X: 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 @ X @ 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 @ X ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8009_gbinomial__partial__sum__poly,axiom,
    ! [M2: nat,A: real,X: 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 @ X @ 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 @ X ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_nat @ M2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M2 ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8010_root__polyfun,axiom,
    ! [N: nat,Z3: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_int @ Z3 @ N )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I: nat] : ( times_times_int @ ( if_int @ ( I = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I = N ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z3 @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_8011_root__polyfun,axiom,
    ! [N: nat,Z3: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_complex @ Z3 @ N )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I: nat] : ( times_times_complex @ ( if_complex @ ( I = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I = N ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z3 @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_8012_root__polyfun,axiom,
    ! [N: nat,Z3: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_8256067586552552935nteger @ Z3 @ N )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I = N ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z3 @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_8013_root__polyfun,axiom,
    ! [N: nat,Z3: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_rat @ Z3 @ N )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I: nat] : ( times_times_rat @ ( if_rat @ ( I = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I = N ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z3 @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_8014_root__polyfun,axiom,
    ! [N: nat,Z3: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N )
     => ( ( ( power_power_real @ Z3 @ N )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I: nat] : ( times_times_real @ ( if_real @ ( I = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I = N ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z3 @ I ) )
            @ ( set_ord_atMost_nat @ N ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_8015_set__decode__plus__power__2,axiom,
    ! [N: nat,Z3: nat] :
      ( ~ ( member_nat @ N @ ( nat_set_decode @ Z3 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ Z3 ) )
        = ( insert_nat @ N @ ( nat_set_decode @ Z3 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_8016_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I ) @ ( semiri8010041392384452111omplex @ I ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8017_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I ) @ ( semiri4939895301339042750nteger @ I ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8018_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I ) @ ( semiri681578069525770553at_rat @ I ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8019_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I ) @ ( semiri1314217659103216013at_int @ I ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8020_choose__alternating__linear__sum,axiom,
    ! [N: nat] :
      ( ( N != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( semiri5074537144036343181t_real @ I ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N @ I ) ) )
          @ ( set_ord_atMost_nat @ N ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8021_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X4: nat] :
          ( collect_nat
          @ ^ [N4: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ) ).

% set_decode_def
thf(fact_8022_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_8023_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_8024_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_8025_binomial__code,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N4 @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N4 @ ( minus_minus_nat @ N4 @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N4 @ K3 ) @ one_one_nat ) @ N4 @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8026_Maclaurin__lemma,axiom,
    ! [H: real,F2: real > real,J2: nat > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ? [B8: real] :
          ( ( F2 @ H )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J2 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
              @ ( set_ord_lessThan_nat @ N ) )
            @ ( times_times_real @ B8 @ ( divide_divide_real @ ( power_power_real @ H @ N ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8027_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > complex > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_8028_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > real > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8029_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > rat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8030_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > nat > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8031_Maclaurin__zero,axiom,
    ! [X: real,N: nat,Diff: nat > int > real] :
      ( ( X = zero_zero_real )
     => ( ( N != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8032_sin__coeff__def,axiom,
    ( sin_coeff
    = ( ^ [N4: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N4 ) ) ) ) ) ).

% sin_coeff_def
thf(fact_8033_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_8034_sin__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
      @ ( sin_real @ X ) ) ).

% sin_paired
thf(fact_8035_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_8036_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_8037_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_8038_bit_Oconj__zero__right,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ X @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_8039_bit_Oconj__zero__left,axiom,
    ! [X: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_8040_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_8041_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_8042_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_8043_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_8044_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8045_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit0 @ X ) ) @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(5)
thf(fact_8046_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_8047_and__numerals_I5_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_8048_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ Y ) ) )
      = zero_z3403309356797280102nteger ) ).

% and_numerals(1)
thf(fact_8049_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_8050_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_8051_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_8052_and__numerals_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_8053_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_8054_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_8055_and__int_Oelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_8056_Maclaurin__sin__expansion3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( sin_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8057_Maclaurin__sin__expansion4,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X )
          & ( ( sin_real @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8058_sumr__cos__zero__one,axiom,
    ! [N: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8059_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_8060_Maclaurin__sin__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( sin_real @ X )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
          @ ( set_ord_lessThan_nat @ N ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_8061_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_8062_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_8063_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_8064_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8065_dbl__inc__simps_I5_J,axiom,
    ! [K2: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K2 ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K2 ) ) ) ).

% dbl_inc_simps(5)
thf(fact_8066_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_8067_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_8068_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_8069_and__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_8070_zdiv__numeral__Bit1,axiom,
    ! [V2: num,W2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_8071_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_8072_and__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_8073_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_8074_sin__npi2,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi2
thf(fact_8075_sin__npi,axiom,
    ! [N: nat] :
      ( ( sin_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% sin_npi
thf(fact_8076_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ one_one_Code_integer )
    = ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8077_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8078_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8079_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_8080_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_8081_and__numerals_I4_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_8082_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_8083_and__numerals_I6_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_8084_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_8085_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_8086_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_8087_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_8088_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_8089_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_8090_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_8091_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_8092_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8093_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_8094_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_8095_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_8096_sin__2pi__minus,axiom,
    ! [X: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( uminus_uminus_real @ ( sin_real @ X ) ) ) ).

% sin_2pi_minus
thf(fact_8097_sin__periodic,axiom,
    ! [X: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X ) ) ).

% sin_periodic
thf(fact_8098_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ ( bit1 @ X ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ Y ) ) )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ ( numera6620942414471956472nteger @ X ) @ ( numera6620942414471956472nteger @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_8099_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_8100_and__numerals_I7_J,axiom,
    ! [X: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_8101_zmod__numeral__Bit1,axiom,
    ! [V2: num,W2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V2 ) ) @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V2 ) @ ( numeral_numeral_int @ W2 ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_8102_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_8103_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_8104_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_8105_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X32: num] :
              ( Y
             != ( bit1 @ X32 ) ) ) ) ).

% num.exhaust
thf(fact_8106_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_Bit1
thf(fact_8107_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_8108_numeral__Bit1,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_Bit1
thf(fact_8109_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_8110_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_8111_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_8112_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_8113_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_8114_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_8115_cong__exp__iff__simps_I13_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q2 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_8116_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_8117_cong__exp__iff__simps_I12_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_8118_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_8119_cong__exp__iff__simps_I10_J,axiom,
    ! [M2: num,Q2: num,N: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_8120_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera6620942414471956472nteger @ ( bit1 @ N ) )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ N ) @ ( numera6620942414471956472nteger @ N ) ) @ one_one_Code_integer ) ) ).

% numeral_code(3)
thf(fact_8121_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_8122_numeral__code_I3_J,axiom,
    ! [N: num] :
      ( ( numera1916890842035813515d_enat @ ( bit1 @ N ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N ) @ ( numera1916890842035813515d_enat @ N ) ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_code(3)
thf(fact_8123_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_8124_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_8125_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_8126_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_8127_power__numeral__odd,axiom,
    ! [Z3: complex,W2: num] :
      ( ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z3 @ ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_complex @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8128_power__numeral__odd,axiom,
    ! [Z3: real,W2: num] :
      ( ( power_power_real @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_real @ ( times_times_real @ Z3 @ ( power_power_real @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_real @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8129_power__numeral__odd,axiom,
    ! [Z3: rat,W2: num] :
      ( ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_rat @ ( times_times_rat @ Z3 @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_rat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8130_power__numeral__odd,axiom,
    ! [Z3: nat,W2: num] :
      ( ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z3 @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_nat @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8131_power__numeral__odd,axiom,
    ! [Z3: int,W2: num] :
      ( ( power_power_int @ Z3 @ ( numeral_numeral_nat @ ( bit1 @ W2 ) ) )
      = ( times_times_int @ ( times_times_int @ Z3 @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) @ ( power_power_int @ Z3 @ ( numeral_numeral_nat @ W2 ) ) ) ) ).

% power_numeral_odd
thf(fact_8132_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_8133_cong__exp__iff__simps_I3_J,axiom,
    ! [N: num,Q2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_8134_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_8135_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_8136_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_8137_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_8138_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_8139_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_8140_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_8141_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_8142_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8143_cong__exp__iff__simps_I7_J,axiom,
    ! [Q2: num,N: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_8144_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ Q2 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_8145_cong__exp__iff__simps_I11_J,axiom,
    ! [M2: num,Q2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q2 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ Q2 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_8146_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_8147_card__3__iff,axiom,
    ! [S3: set_complex] :
      ( ( ( finite_card_complex @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: complex,Y4: complex,Z4: complex] :
            ( ( S3
              = ( insert_complex @ X4 @ ( insert_complex @ Y4 @ ( insert_complex @ Z4 @ bot_bot_set_complex ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8148_card__3__iff,axiom,
    ! [S3: set_list_nat] :
      ( ( ( finite_card_list_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: list_nat,Y4: list_nat,Z4: list_nat] :
            ( ( S3
              = ( insert_list_nat @ X4 @ ( insert_list_nat @ Y4 @ ( insert_list_nat @ Z4 @ bot_bot_set_list_nat ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8149_card__3__iff,axiom,
    ! [S3: set_set_nat] :
      ( ( ( finite_card_set_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: set_nat,Y4: set_nat,Z4: set_nat] :
            ( ( S3
              = ( insert_set_nat @ X4 @ ( insert_set_nat @ Y4 @ ( insert_set_nat @ Z4 @ bot_bot_set_set_nat ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8150_card__3__iff,axiom,
    ! [S3: set_nat] :
      ( ( ( finite_card_nat @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: nat,Y4: nat,Z4: nat] :
            ( ( S3
              = ( insert_nat @ X4 @ ( insert_nat @ Y4 @ ( insert_nat @ Z4 @ bot_bot_set_nat ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8151_card__3__iff,axiom,
    ! [S3: set_int] :
      ( ( ( finite_card_int @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: int,Y4: int,Z4: int] :
            ( ( S3
              = ( insert_int @ X4 @ ( insert_int @ Y4 @ ( insert_int @ Z4 @ bot_bot_set_int ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8152_card__3__iff,axiom,
    ! [S3: set_real] :
      ( ( ( finite_card_real @ S3 )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( ? [X4: real,Y4: real,Z4: real] :
            ( ( S3
              = ( insert_real @ X4 @ ( insert_real @ Y4 @ ( insert_real @ Z4 @ bot_bot_set_real ) ) ) )
            & ( X4 != Y4 )
            & ( Y4 != Z4 )
            & ( X4 != Z4 ) ) ) ) ).

% card_3_iff
thf(fact_8153_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_8154_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_8155_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_8156_sin__pi__divide__n__ge__0,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% sin_pi_divide_n_ge_0
thf(fact_8157_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_8158_sin__lt__zero,axiom,
    ! [X: real] :
      ( ( ord_less_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8159_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N4: nat] :
          ( if_nat
          @ ( ( M6 = zero_zero_nat )
            | ( N4 = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_8160_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_8161_sin__le__zero,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ pi @ X )
     => ( ( ord_less_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8162_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_8163_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_8164_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_8165_sin__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( sin_real @ X )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_8166_sin__zero__iff,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_8167_Maclaurin__minus__cos__expansion,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8168_Maclaurin__cos__expansion2,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X )
            & ( ( cos_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8169_Maclaurin__sin__expansion2,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( sin_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_8170_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_8171_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_8172_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_8173_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_8174_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_8175_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_8176_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_8177_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_8178_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_8179_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_8180_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_8181_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_8182_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_8183_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_8184_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_8185_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_8186_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_8187_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_8188_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_8189_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_8190_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_8191_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_8192_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_8193_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_8194_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_8195_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_8196_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_8197_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_8198_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_8199_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_8200_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_8201_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_8202_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_numeral
thf(fact_8203_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N ) )
      = ( numeral_numeral_rat @ N ) ) ).

% abs_numeral
thf(fact_8204_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N ) )
      = ( numeral_numeral_real @ N ) ) ).

% abs_numeral
thf(fact_8205_abs__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N ) )
      = ( numeral_numeral_int @ N ) ) ).

% abs_numeral
thf(fact_8206_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_8207_abs__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
      = ( times_times_real @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_8208_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_8209_abs__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
      = ( times_times_int @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_8210_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_8211_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_8212_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_8213_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_8214_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_8215_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_8216_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_8217_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_8218_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_8219_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_8220_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_8221_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_8222_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_8223_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_8224_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_8225_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_8226_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_8227_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_8228_dvd__abs__iff,axiom,
    ! [M2: real,K2: real] :
      ( ( dvd_dvd_real @ M2 @ ( abs_abs_real @ K2 ) )
      = ( dvd_dvd_real @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_8229_dvd__abs__iff,axiom,
    ! [M2: int,K2: int] :
      ( ( dvd_dvd_int @ M2 @ ( abs_abs_int @ K2 ) )
      = ( dvd_dvd_int @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_8230_dvd__abs__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( ( dvd_dvd_Code_integer @ M2 @ ( abs_abs_Code_integer @ K2 ) )
      = ( dvd_dvd_Code_integer @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_8231_dvd__abs__iff,axiom,
    ! [M2: rat,K2: rat] :
      ( ( dvd_dvd_rat @ M2 @ ( abs_abs_rat @ K2 ) )
      = ( dvd_dvd_rat @ M2 @ K2 ) ) ).

% dvd_abs_iff
thf(fact_8232_abs__dvd__iff,axiom,
    ! [M2: real,K2: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M2 ) @ K2 )
      = ( dvd_dvd_real @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_8233_abs__dvd__iff,axiom,
    ! [M2: int,K2: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M2 ) @ K2 )
      = ( dvd_dvd_int @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_8234_abs__dvd__iff,axiom,
    ! [M2: code_integer,K2: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M2 ) @ K2 )
      = ( dvd_dvd_Code_integer @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_8235_abs__dvd__iff,axiom,
    ! [M2: rat,K2: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M2 ) @ K2 )
      = ( dvd_dvd_rat @ M2 @ K2 ) ) ).

% abs_dvd_iff
thf(fact_8236_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N ) )
      = ( semiri4939895301339042750nteger @ N ) ) ).

% abs_of_nat
thf(fact_8237_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N ) )
      = ( semiri681578069525770553at_rat @ N ) ) ).

% abs_of_nat
thf(fact_8238_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N ) )
      = ( semiri5074537144036343181t_real @ N ) ) ).

% abs_of_nat
thf(fact_8239_abs__of__nat,axiom,
    ! [N: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
      = ( semiri1314217659103216013at_int @ N ) ) ).

% abs_of_nat
thf(fact_8240_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( zero_n3304061248610475627l_real @ P2 ) ) ).

% abs_bool_eq
thf(fact_8241_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( zero_n2052037380579107095ol_rat @ P2 ) ) ).

% abs_bool_eq
thf(fact_8242_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( zero_n2684676970156552555ol_int @ P2 ) ) ).

% abs_bool_eq
thf(fact_8243_abs__bool__eq,axiom,
    ! [P2: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( zero_n356916108424825756nteger @ P2 ) ) ).

% abs_bool_eq
thf(fact_8244_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_8245_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_8246_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_8247_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_8248_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_8249_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_8250_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_8251_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_8252_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_8253_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_8254_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_8255_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_8256_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_8257_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_8258_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_8259_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_8260_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_8261_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_8262_abs__neg__numeral,axiom,
    ! [N: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N ) ) )
      = ( numera6620942414471956472nteger @ N ) ) ).

% abs_neg_numeral
thf(fact_8263_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_8264_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_8265_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_8266_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_8267_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_8268_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_8269_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_8270_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_8271_eq__numeral__Suc,axiom,
    ! [K2: num,N: nat] :
      ( ( ( numeral_numeral_nat @ K2 )
        = ( suc @ N ) )
      = ( ( pred_numeral @ K2 )
        = N ) ) ).

% eq_numeral_Suc
thf(fact_8272_Suc__eq__numeral,axiom,
    ! [N: nat,K2: num] :
      ( ( ( suc @ N )
        = ( numeral_numeral_nat @ K2 ) )
      = ( N
        = ( pred_numeral @ K2 ) ) ) ).

% Suc_eq_numeral
thf(fact_8273_sum__abs,axiom,
    ! [F2: int > int,A4: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F2 @ A4 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( abs_abs_int @ ( F2 @ I ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_8274_sum__abs,axiom,
    ! [F2: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F2 @ A4 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( abs_abs_real @ ( F2 @ I ) )
        @ A4 ) ) ).

% sum_abs
thf(fact_8275_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_8276_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_8277_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_8278_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_8279_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_8280_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_8281_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_8282_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_8283_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_8284_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_8285_pred__numeral__simps_I3_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit1 @ K2 ) )
      = ( numeral_numeral_nat @ ( bit0 @ K2 ) ) ) ).

% pred_numeral_simps(3)
thf(fact_8286_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_8287_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_8288_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_8289_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_8290_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_8291_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_8292_sum__abs__ge__zero,axiom,
    ! [F2: int > int,A4: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I: int] : ( abs_abs_int @ ( F2 @ I ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_8293_sum__abs__ge__zero,axiom,
    ! [F2: nat > real,A4: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I: nat] : ( abs_abs_real @ ( F2 @ I ) )
        @ A4 ) ) ).

% sum_abs_ge_zero
thf(fact_8294_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_8295_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_8296_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_8297_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_8298_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_8299_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_8300_sin__cos__squared__add3,axiom,
    ! [X: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X ) @ ( cos_complex @ X ) ) @ ( times_times_complex @ ( sin_complex @ X ) @ ( sin_complex @ X ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_8301_sin__cos__squared__add3,axiom,
    ! [X: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ X ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ X ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_8302_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_8303_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_8304_divmod__algorithm__code_I2_J,axiom,
    ! [M2: num] :
      ( ( unique3479559517661332726nteger @ M2 @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M2 ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_8305_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_8306_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_8307_divmod__algorithm__code_I3_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_8308_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_8309_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_8310_divmod__algorithm__code_I4_J,axiom,
    ! [N: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_8311_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_8312_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_8313_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_8314_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_8315_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_8316_cos__periodic,axiom,
    ! [X: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X ) ) ).

% cos_periodic
thf(fact_8317_cos__2pi__minus,axiom,
    ! [X: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X ) )
      = ( cos_real @ X ) ) ).

% cos_2pi_minus
thf(fact_8318_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_8319_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_8320_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_8321_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_8322_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_8323_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_8324_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_8325_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_8326_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_8327_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_8328_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_8329_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_8330_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_8331_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_8332_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_8333_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_8334_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_8335_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_8336_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_8337_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_8338_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_8339_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_8340_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_8341_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_8342_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_8343_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_8344_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_8345_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_8346_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_8347_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_8348_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_8349_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_8350_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_8351_abs__eq__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ( abs_abs_int @ X )
        = ( abs_abs_int @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8352_abs__eq__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ( abs_abs_real @ X )
        = ( abs_abs_real @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8353_abs__eq__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8354_abs__eq__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X )
        = ( abs_abs_rat @ Y ) )
      = ( ( X = Y )
        | ( X
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_8355_dvd__if__abs__eq,axiom,
    ! [L: real,K2: real] :
      ( ( ( abs_abs_real @ L )
        = ( abs_abs_real @ K2 ) )
     => ( dvd_dvd_real @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_8356_dvd__if__abs__eq,axiom,
    ! [L: int,K2: int] :
      ( ( ( abs_abs_int @ L )
        = ( abs_abs_int @ K2 ) )
     => ( dvd_dvd_int @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_8357_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K2: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K2 ) )
     => ( dvd_dvd_Code_integer @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_8358_dvd__if__abs__eq,axiom,
    ! [L: rat,K2: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K2 ) )
     => ( dvd_dvd_rat @ L @ K2 ) ) ).

% dvd_if_abs_eq
thf(fact_8359_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_8360_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_8361_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_8362_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_8363_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_8364_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_8365_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_8366_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_8367_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_8368_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_8369_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_8370_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_8371_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_8372_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_8373_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_8374_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_8375_abs__mult__less,axiom,
    ! [A: code_integer,C2: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C2 )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C2 @ D ) ) ) ) ).

% abs_mult_less
thf(fact_8376_abs__mult__less,axiom,
    ! [A: real,C2: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C2 )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C2 @ D ) ) ) ) ).

% abs_mult_less
thf(fact_8377_abs__mult__less,axiom,
    ! [A: rat,C2: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C2 )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C2 @ D ) ) ) ) ).

% abs_mult_less
thf(fact_8378_abs__mult__less,axiom,
    ! [A: int,C2: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C2 )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C2 @ D ) ) ) ) ).

% abs_mult_less
thf(fact_8379_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_8380_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_8381_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_8382_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_8383_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_8384_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_8385_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_8386_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_8387_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_8388_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_8389_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_8390_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_8391_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_8392_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_8393_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_8394_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_8395_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_8396_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_8397_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_8398_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_8399_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_8400_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_8401_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_8402_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_8403_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_8404_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_8405_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_8406_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_8407_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_8408_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_8409_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_8410_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_8411_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_8412_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_8413_polar__Ex,axiom,
    ! [X: real,Y: real] :
    ? [R4: real,A3: real] :
      ( ( X
        = ( times_times_real @ R4 @ ( cos_real @ A3 ) ) )
      & ( Y
        = ( times_times_real @ R4 @ ( sin_real @ A3 ) ) ) ) ).

% polar_Ex
thf(fact_8414_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_8415_sin__cos__le1,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_8416_dense__eq0__I,axiom,
    ! [X: real] :
      ( ! [E2: real] :
          ( ( ord_less_real @ zero_zero_real @ E2 )
         => ( ord_less_eq_real @ ( abs_abs_real @ X ) @ E2 ) )
     => ( X = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_8417_dense__eq0__I,axiom,
    ! [X: rat] :
      ( ! [E2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E2 )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ E2 ) )
     => ( X = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_8418_abs__mult__pos,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_8419_abs__mult__pos,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X )
        = ( abs_abs_real @ ( times_times_real @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_8420_abs__mult__pos,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_8421_abs__mult__pos,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X )
        = ( abs_abs_int @ ( times_times_int @ Y @ X ) ) ) ) ).

% abs_mult_pos
thf(fact_8422_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_8423_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_8424_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_8425_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_8426_cos__one__sin__zero,axiom,
    ! [X: complex] :
      ( ( ( cos_complex @ X )
        = one_one_complex )
     => ( ( sin_complex @ X )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_8427_cos__one__sin__zero,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
     => ( ( sin_real @ X )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_8428_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_8429_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_8430_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_8431_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_8432_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_8433_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_8434_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_8435_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_8436_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_8437_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_8438_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_8439_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_8440_abs__div__pos,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_8441_abs__div__pos,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_8442_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_8443_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_8444_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_8445_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_8446_abs__if__raw,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_8447_abs__if__raw,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_8448_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_8449_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if_raw
thf(fact_8450_abs__if,axiom,
    ( abs_abs_int
    = ( ^ [A5: int] : ( if_int @ ( ord_less_int @ A5 @ zero_zero_int ) @ ( uminus_uminus_int @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_8451_abs__if,axiom,
    ( abs_abs_real
    = ( ^ [A5: real] : ( if_real @ ( ord_less_real @ A5 @ zero_zero_real ) @ ( uminus_uminus_real @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_8452_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A5: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A5 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_8453_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A5: rat] : ( if_rat @ ( ord_less_rat @ A5 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A5 ) @ A5 ) ) ) ).

% abs_if
thf(fact_8454_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_8455_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_8456_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_8457_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_8458_sin__add,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_8459_abs__diff__le__iff,axiom,
    ! [X: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R3 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X )
        & ( ord_le3102999989581377725nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8460_abs__diff__le__iff,axiom,
    ! [X: real,A: real,R3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R3 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R3 ) @ X )
        & ( ord_less_eq_real @ X @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8461_abs__diff__le__iff,axiom,
    ! [X: rat,A: rat,R3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R3 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R3 ) @ X )
        & ( ord_less_eq_rat @ X @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8462_abs__diff__le__iff,axiom,
    ! [X: int,A: int,R3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R3 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R3 ) @ X )
        & ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_8463_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C2: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C2 @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C2 ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8464_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C2: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C2 @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C2 ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8465_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C2: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C2 @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C2 ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8466_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C2: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_8467_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_8468_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_8469_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_8470_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_8471_sin__diff,axiom,
    ! [X: real,Y: real] :
      ( ( sin_real @ ( minus_minus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( sin_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% sin_diff
thf(fact_8472_abs__diff__less__iff,axiom,
    ! [X: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X @ A ) ) @ R3 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X )
        & ( ord_le6747313008572928689nteger @ X @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8473_abs__diff__less__iff,axiom,
    ! [X: real,A: real,R3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X @ A ) ) @ R3 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R3 ) @ X )
        & ( ord_less_real @ X @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8474_abs__diff__less__iff,axiom,
    ! [X: rat,A: rat,R3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X @ A ) ) @ R3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R3 ) @ X )
        & ( ord_less_rat @ X @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8475_abs__diff__less__iff,axiom,
    ! [X: int,A: int,R3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R3 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R3 ) @ X )
        & ( ord_less_int @ X @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_8476_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_8477_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_8478_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_8479_cos__add,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_8480_cos__diff,axiom,
    ! [X: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_8481_abs__add__one__gt__zero,axiom,
    ! [X: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_8482_abs__add__one__gt__zero,axiom,
    ! [X: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_8483_abs__add__one__gt__zero,axiom,
    ! [X: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_8484_abs__add__one__gt__zero,axiom,
    ! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).

% abs_add_one_gt_zero
thf(fact_8485_sin__zero__norm__cos__one,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8486_sin__zero__norm__cos__one,axiom,
    ! [X: complex] :
      ( ( ( sin_complex @ X )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8487_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_8488_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri4449623510593786356d_enat @ ( numeral_numeral_nat @ K2 ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ K2 ) @ ( semiri4449623510593786356d_enat @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8489_fact__numeral,axiom,
    ! [K2: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K2 ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K2 ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K2 ) ) ) ) ).

% fact_numeral
thf(fact_8490_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_8491_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_8492_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_8493_sin__double,axiom,
    ! [X: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X ) ) @ ( cos_complex @ X ) ) ) ).

% sin_double
thf(fact_8494_sin__double,axiom,
    ! [X: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X ) ) @ ( cos_real @ X ) ) ) ).

% sin_double
thf(fact_8495_abs__le__square__iff,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_8496_abs__le__square__iff,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_8497_abs__le__square__iff,axiom,
    ! [X: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_8498_abs__le__square__iff,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_8499_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_int_def
thf(fact_8500_abs__sqrt__wlog,axiom,
    ! [P2: code_integer > code_integer > $o,X: code_integer] :
      ( ! [X3: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
         => ( P2 @ X3 @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_Code_integer @ X ) @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8501_abs__sqrt__wlog,axiom,
    ! [P2: real > real > $o,X: real] :
      ( ! [X3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( P2 @ X3 @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_real @ X ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8502_abs__sqrt__wlog,axiom,
    ! [P2: rat > rat > $o,X: rat] :
      ( ! [X3: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
         => ( P2 @ X3 @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_rat @ X ) @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8503_abs__sqrt__wlog,axiom,
    ! [P2: int > int > $o,X: int] :
      ( ! [X3: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( P2 @ X3 @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P2 @ ( abs_abs_int @ X ) @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_8504_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_8505_power2__le__iff__abs__le,axiom,
    ! [Y: real,X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_8506_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_8507_power2__le__iff__abs__le,axiom,
    ! [Y: int,X: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_8508_abs__square__le__1,axiom,
    ! [X: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_8509_abs__square__le__1,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_8510_abs__square__le__1,axiom,
    ! [X: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_8511_abs__square__le__1,axiom,
    ! [X: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_8512_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N4: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N4 ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_8513_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N4: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N4 ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_8514_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod_def
thf(fact_8515_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N4: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N4 ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N4 ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_8516_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_8517_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_8518_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_8519_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_8520_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X @ 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
                  @ ^ [I: complex] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8521_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X @ 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
                  @ ^ [I: real] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8522_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X @ 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
                  @ ^ [I: nat] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8523_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X @ I3 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X @ 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
                  @ ^ [I: int] : ( times_3573771949741848930nteger @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8524_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X @ 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
                  @ ^ [I: complex] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8525_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X @ 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
                  @ ^ [I: real] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8526_convex__sum__bound__le,axiom,
    ! [I5: set_int,X: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X @ I3 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X @ 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
                  @ ^ [I: int] : ( times_times_real @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8527_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I3: complex] :
          ( ( member_complex @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X @ 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
                  @ ^ [I: complex] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8528_convex__sum__bound__le,axiom,
    ! [I5: set_real,X: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: real] :
              ( ( member_real @ 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
                @ ( groups1300246762558778688al_rat
                  @ ^ [I: real] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8529_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X @ I3 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X @ I5 )
          = one_one_rat )
       => ( ! [I3: nat] :
              ( ( member_nat @ 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
                @ ( groups2906978787729119204at_rat
                  @ ^ [I: nat] : ( times_times_rat @ ( A @ I ) @ ( X @ I ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_8530_cos__plus__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8531_cos__plus__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8532_cos__times__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8533_cos__times__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z3 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8534_cos__double__less__one,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8535_cos__double__cos,axiom,
    ! [W2: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_8536_cos__double__cos,axiom,
    ! [W2: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W2 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_8537_cos__treble__cos,axiom,
    ! [X: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8538_cos__treble__cos,axiom,
    ! [X: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X ) ) ) ) ).

% cos_treble_cos
thf(fact_8539_cos__diff__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z3 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8540_cos__diff__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z3 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8541_sin__diff__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8542_sin__diff__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8543_sin__plus__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8544_sin__plus__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8545_cos__times__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8546_cos__times__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z3 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8547_sin__times__cos,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8548_sin__times__cos,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z3 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z3 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8549_sin__times__sin,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z3 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z3 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z3 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8550_sin__times__sin,axiom,
    ! [W2: real,Z3: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z3 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z3 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8551_Maclaurin__cos__expansion,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( cos_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T6 @ ( times_times_real @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ N ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_8552_cos__one__2pi,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X4: nat] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X4: nat] :
            ( X
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_8553_sin__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8554_cos__zero__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ( cos_real @ X )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_8555_cos__zero__iff,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N4: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 )
            & ( X
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_8556_cos__expansion__lemma,axiom,
    ! [X: real,M2: nat] :
      ( ( cos_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8557_cos__paired,axiom,
    ! [X: real] :
      ( sums_real
      @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_real @ X @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
      @ ( cos_real @ X ) ) ).

% cos_paired
thf(fact_8558_sincos__total__2pi__le,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X
            = ( cos_real @ T6 ) )
          & ( Y
            = ( sin_real @ T6 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_8559_sincos__total__2pi,axiom,
    ! [X: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X
                  = ( cos_real @ T6 ) )
               => ( Y
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8560_monoseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( topolo6980174941875973593q_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8561_summable__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8562_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
     => ( ( ord_less_eq_real @ X @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_8563_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_8564_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_8565_tan__npi,axiom,
    ! [N: nat] :
      ( ( tan_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% tan_npi
thf(fact_8566_tan__periodic__n,axiom,
    ! [X: real,N: num] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_n
thf(fact_8567_tan__periodic__nat,axiom,
    ! [X: real,N: nat] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_nat
thf(fact_8568_tan__periodic,axiom,
    ! [X: real] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic
thf(fact_8569_abs__zmult__eq__1,axiom,
    ! [M2: int,N: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M2 @ N ) )
        = one_one_int )
     => ( ( abs_abs_int @ M2 )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_8570_abs__div,axiom,
    ! [Y: int,X: int] :
      ( ( dvd_dvd_int @ Y @ X )
     => ( ( abs_abs_int @ ( divide_divide_int @ X @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_8571_summable__rabs__comparison__test,axiom,
    ! [F2: nat > real,G2: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( abs_abs_real @ ( F2 @ N3 ) ) @ ( G2 @ N3 ) ) )
     => ( ( summable_real @ G2 )
       => ( summable_real
          @ ^ [N4: nat] : ( abs_abs_real @ ( F2 @ N4 ) ) ) ) ) ).

% summable_rabs_comparison_test
thf(fact_8572_ln__mult,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_8573_zdvd__mult__cancel1,axiom,
    ! [M2: int,N: int] :
      ( ( M2 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M2 @ N ) @ M2 )
        = ( ( abs_abs_int @ N )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_8574_sum__pos__lt__pair,axiom,
    ! [F2: nat > real,K2: nat] :
      ( ( summable_real @ F2 )
     => ( ! [D5: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F2 @ ( plus_plus_nat @ K2 @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) ) ) @ ( F2 @ ( plus_plus_nat @ K2 @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D5 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F2 @ ( set_ord_lessThan_nat @ K2 ) ) @ ( suminf_real @ F2 ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8575_ln__realpow,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ln_ln_real @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( ln_ln_real @ X ) ) ) ) ).

% ln_realpow
thf(fact_8576_summable__power__series,axiom,
    ! [F2: nat > real,Z3: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F2 @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F2 @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( summable_real
              @ ^ [I: nat] : ( times_times_real @ ( F2 @ I ) @ ( power_power_real @ Z3 @ I ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8577_nat__intermed__int__val,axiom,
    ! [M2: nat,N: nat,F2: 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 @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M2 @ N )
       => ( ( ord_less_eq_int @ ( F2 @ M2 ) @ K2 )
         => ( ( ord_less_eq_int @ K2 @ ( F2 @ N ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M2 @ I3 )
                & ( ord_less_eq_nat @ I3 @ N )
                & ( ( F2 @ I3 )
                  = K2 ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_8578_incr__lemma,axiom,
    ! [D: int,Z3: int,X: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z3 @ ( plus_plus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_8579_decr__lemma,axiom,
    ! [D: int,X: int,Z3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z3 ) ) @ one_one_int ) @ D ) ) @ Z3 ) ) ).

% decr_lemma
thf(fact_8580_ln__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( ord_less_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X )
          = ( suminf_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X @ one_one_real ) @ ( suc @ N4 ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8581_nat__ivt__aux,axiom,
    ! [N: nat,F2: nat > int,K2: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F2 @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F2 @ I3 )
                = K2 ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_8582_nat0__intermed__int__val,axiom,
    ! [N: nat,F2: nat > int,K2: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F2 @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F2 @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F2 @ zero_zero_nat ) @ K2 )
       => ( ( ord_less_eq_int @ K2 @ ( F2 @ N ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N )
              & ( ( F2 @ I3 )
                = K2 ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_8583_ln__one__minus__pos__lower__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_8584_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X ) ) @ X ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_8585_and__int_Opelims,axiom,
    ! [X: int,Xa2: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X @ Xa2 )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_8586_arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( arctan @ X )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8587_Maclaurin__exp__lt,axiom,
    ! [X: real,N: nat] :
      ( ( X != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
            & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( exp_real @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8588_arctan__double,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_8589_real__exp__bound__lemma,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_8590_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ X )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ X ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8591_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X: real,N: nat] :
      ( ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ N ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X @ ( semiri5074537144036343181t_real @ N ) ) ) @ N ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8592_arctan__add,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_8593_Maclaurin__exp__le,axiom,
    ! [X: real,N: nat] :
    ? [T6: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
      & ( ( exp_real @ X )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_8594_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_8595_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_8596_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_8597_Maclaurin__sin__bound,axiom,
    ! [X: real,N: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X )
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X @ M6 ) )
            @ ( set_ord_lessThan_nat @ N ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( abs_abs_real @ X ) @ N ) ) ) ).

% Maclaurin_sin_bound
thf(fact_8598_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_8599_pred__numeral__simps_I2_J,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( bit0 @ K2 ) )
      = ( numeral_numeral_nat @ ( bitM @ K2 ) ) ) ).

% pred_numeral_simps(2)
thf(fact_8600_sin__npi__int,axiom,
    ! [N: int] :
      ( ( sin_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N ) ) )
      = zero_zero_real ) ).

% sin_npi_int
thf(fact_8601_tan__periodic__int,axiom,
    ! [X: real,I2: int] :
      ( ( tan_real @ ( plus_plus_real @ X @ ( times_times_real @ ( ring_1_of_int_real @ I2 ) @ pi ) ) )
      = ( tan_real @ X ) ) ).

% tan_periodic_int
thf(fact_8602_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_8603_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_8604_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_8605_real__scaleR__def,axiom,
    real_V1485227260804924795R_real = times_times_real ).

% real_scaleR_def
thf(fact_8606_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X4: real,Y4: real] : ( times_times_real @ X4 @ ( inverse_inverse_real @ Y4 ) ) ) ) ).

% divide_real_def
thf(fact_8607_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_8608_BitM__plus__one,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ ( bitM @ N ) @ one )
      = ( bit0 @ N ) ) ).

% BitM_plus_one
thf(fact_8609_one__plus__BitM,axiom,
    ! [N: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% one_plus_BitM
thf(fact_8610_forall__pos__mono__1,axiom,
    ! [P2: real > $o,E: real] :
      ( ! [D5: real,E2: real] :
          ( ( ord_less_real @ D5 @ E2 )
         => ( ( P2 @ D5 )
           => ( P2 @ E2 ) ) )
     => ( ! [N3: nat] : ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P2 @ E ) ) ) ) ).

% forall_pos_mono_1
thf(fact_8611_forall__pos__mono,axiom,
    ! [P2: real > $o,E: real] :
      ( ! [D5: real,E2: real] :
          ( ( ord_less_real @ D5 @ E2 )
         => ( ( P2 @ D5 )
           => ( P2 @ E2 ) ) )
     => ( ! [N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( P2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
       => ( ( ord_less_real @ zero_zero_real @ E )
         => ( P2 @ E ) ) ) ) ).

% forall_pos_mono
thf(fact_8612_real__arch__inverse,axiom,
    ! [E: real] :
      ( ( ord_less_real @ zero_zero_real @ E )
      = ( ? [N4: nat] :
            ( ( N4 != zero_zero_nat )
            & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) )
            & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N4 ) ) @ E ) ) ) ) ).

% real_arch_inverse
thf(fact_8613_sin__zero__iff__int2,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( X
            = ( times_times_real @ ( ring_1_of_int_real @ I ) @ pi ) ) ) ) ).

% sin_zero_iff_int2
thf(fact_8614_cos__one__2pi__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = one_one_real )
      = ( ? [X4: int] :
            ( X
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_8615_cos__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( cos_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_8616_sin__zero__iff__int,axiom,
    ! [X: real] :
      ( ( ( sin_real @ X )
        = zero_zero_real )
      = ( ? [I: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I )
            & ( X
              = ( times_times_real @ ( ring_1_of_int_real @ I ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_8617_complex__unimodular__polar,axiom,
    ! [Z3: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z3 )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z3
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8618_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_8619_log__base__10__eq1,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq1
thf(fact_8620_complex__scaleR,axiom,
    ! [R3: real,A: real,B: real] :
      ( ( real_V2046097035970521341omplex @ R3 @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( times_times_real @ R3 @ A ) @ ( times_times_real @ R3 @ B ) ) ) ).

% complex_scaleR
thf(fact_8621_complex__mult,axiom,
    ! [A: real,B: real,C2: real,D: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ ( complex2 @ C2 @ D ) )
      = ( complex2 @ ( minus_minus_real @ ( times_times_real @ A @ C2 ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C2 ) ) ) ) ).

% complex_mult
thf(fact_8622_log__mult,axiom,
    ! [A: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log2 @ A @ ( times_times_real @ X @ Y ) )
              = ( plus_plus_real @ ( log2 @ A @ X ) @ ( log2 @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8623_log__nat__power,axiom,
    ! [X: real,B: real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log2 @ B @ ( power_power_real @ X @ N ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ).

% log_nat_power
thf(fact_8624_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 @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_less
thf(fact_8625_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( ( log2 @ A @ X )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log2 @ B @ X ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8626_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 @ ( log2 @ B @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_of_power_le
thf(fact_8627_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 ) @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% le_log2_of_power
thf(fact_8628_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_8629_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 @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_less
thf(fact_8630_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_8631_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 @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ).

% log2_of_power_le
thf(fact_8632_log__base__10__eq2,axiom,
    ! [X: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X )
        = ( times_times_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X ) ) ) ) ).

% log_base_10_eq2
thf(fact_8633_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 @ ( log2 @ ( 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_8634_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 @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8635_ceiling__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim7802044766580827645g_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log2 @ ( 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_8636_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N4: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N4 = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N4 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N4 ) @ N4 ) ) ) ) ) ).

% divmod_nat_if
thf(fact_8637_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 @ ( log2 @ ( 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_8638_divide__complex__def,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X4: complex,Y4: complex] : ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y4 ) ) ) ) ).

% divide_complex_def
thf(fact_8639_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M6: nat,N4: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N4 ) ) @ M6 ) ) ) ).

% prod_encode_def
thf(fact_8640_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q5: int,R: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_8641_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N4: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N4 ) @ ( modulo_modulo_nat @ M6 @ N4 ) ) ) ) ).

% divmod_nat_def
thf(fact_8642_floor__log2__div2,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N ) ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).

% floor_log2_div2
thf(fact_8643_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 @ ( log2 @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K2 ) ) )
            = ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8644_cot__periodic,axiom,
    ! [X: real] :
      ( ( cot_real @ ( plus_plus_real @ X @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X ) ) ).

% cot_periodic
thf(fact_8645_modulo__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 ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) )
      & ( ~ ( ( ( 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 ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K2 )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N @ M2 ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_8646_dvd__mult__sgn__iff,axiom,
    ! [L: int,K2: int,R3: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ K2 @ ( sgn_sgn_int @ R3 ) ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_mult_sgn_iff
thf(fact_8647_dvd__sgn__mult__iff,axiom,
    ! [L: int,R3: int,K2: int] :
      ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ K2 ) )
      = ( ( dvd_dvd_int @ L @ K2 )
        | ( R3 = zero_zero_int ) ) ) ).

% dvd_sgn_mult_iff
thf(fact_8648_mult__sgn__dvd__iff,axiom,
    ! [L: int,R3: int,K2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R3 ) ) @ K2 )
      = ( ( dvd_dvd_int @ L @ K2 )
        & ( ( R3 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% mult_sgn_dvd_iff
thf(fact_8649_sgn__mult__dvd__iff,axiom,
    ! [R3: int,L: int,K2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R3 ) @ L ) @ K2 )
      = ( ( dvd_dvd_int @ L @ K2 )
        & ( ( R3 = zero_zero_int )
         => ( K2 = zero_zero_int ) ) ) ) ).

% sgn_mult_dvd_iff
thf(fact_8650_cot__npi,axiom,
    ! [N: nat] :
      ( ( cot_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ pi ) )
      = zero_zero_real ) ).

% cot_npi
thf(fact_8651_int__sgnE,axiom,
    ! [K2: int] :
      ~ ! [N3: nat,L4: int] :
          ( K2
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_8652_div__eq__sgn__abs,axiom,
    ! [K2: int,L: int] :
      ( ( ( sgn_sgn_int @ K2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K2 @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_8653_div__sgn__abs__cancel,axiom,
    ! [V2: int,K2: int,L: int] :
      ( ( V2 != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V2 ) @ ( abs_abs_int @ K2 ) ) @ ( times_times_int @ ( sgn_sgn_int @ V2 ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_8654_div__dvd__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( dvd_dvd_int @ L @ K2 )
     => ( ( divide_divide_int @ K2 @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K2 ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_8655_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K2: int,Q2: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
       => ( ( K2
            = ( plus_plus_int @ ( times_times_int @ Q2 @ L ) @ R3 ) )
         => ( eucl_rel_int @ K2 @ L @ ( product_Pair_int_int @ Q2 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_8656_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A13: int,A24: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A13 = K3 )
              & ( A24 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q5: int] :
              ( ( A13 = K3 )
              & ( A24 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q5 @ L2 ) ) )
          | ? [R: int,L2: int,K3: int,Q5: int] :
              ( ( A13 = K3 )
              & ( A24 = L2 )
              & ( A32
                = ( product_Pair_int_int @ Q5 @ R ) )
              & ( ( sgn_sgn_int @ R )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_8657_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q3: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q3 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q3 @ A23 ) ) ) )
         => ~ ! [R4: int,Q3: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q3 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q3 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_8658_div__noneq__sgn__abs,axiom,
    ! [L: int,K2: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K2 )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K2 @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K2 ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_8659_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_8660_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_8661_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_8662_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_8663_mask__nat__positive__iff,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% mask_nat_positive_iff
thf(fact_8664_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_8665_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( ( nat2 @ I2 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_8666_nat__le__0,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ Z3 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_8667_nat__neg__numeral,axiom,
    ! [K2: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K2 ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_8668_nat__zminus__int,axiom,
    ! [N: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_8669_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_8670_zero__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ zero_zero_int @ Z3 ) ) ).

% zero_less_nat_eq
thf(fact_8671_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_8672_nat__ceiling__le__eq,axiom,
    ! [X: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X ) ) @ A )
      = ( ord_less_eq_real @ X @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_8673_one__less__nat__eq,axiom,
    ! [Z3: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z3 ) )
      = ( ord_less_int @ one_one_int @ Z3 ) ) ).

% one_less_nat_eq
thf(fact_8674_numeral__power__le__nat__cancel__iff,axiom,
    ! [X: num,N: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_8675_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X: num,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X ) @ N ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X ) @ N ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_8676_take__bit__tightened__less__eq__nat,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M2 @ Q2 ) @ ( bit_se2925701944663578781it_nat @ N @ Q2 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_8677_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_8678_take__bit__mult,axiom,
    ! [N: nat,K2: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N @ K2 ) @ ( bit_se2923211474154528505it_int @ N @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N @ ( times_times_int @ K2 @ L ) ) ) ).

% take_bit_mult
thf(fact_8679_less__eq__mask,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ ( bit_se2002935070580805687sk_nat @ N ) ) ).

% less_eq_mask
thf(fact_8680_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_8681_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_8682_nat__mono,axiom,
    ! [X: int,Y: int] :
      ( ( ord_less_eq_int @ X @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_8683_nat__le__iff,axiom,
    ! [X: int,N: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
      = ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).

% nat_le_iff
thf(fact_8684_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z3: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z3 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z3 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_8685_nat__times__as__int,axiom,
    ( times_times_nat
    = ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).

% nat_times_as_int
thf(fact_8686_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_8687_nat__le__eq__zle,axiom,
    ! [W2: int,Z3: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z3 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z3 ) )
        = ( ord_less_eq_int @ W2 @ Z3 ) ) ) ).

% nat_le_eq_zle
thf(fact_8688_nat__eq__iff2,axiom,
    ! [M2: nat,W2: int] :
      ( ( M2
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_8689_nat__eq__iff,axiom,
    ! [W2: int,M2: nat] :
      ( ( ( nat2 @ W2 )
        = M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_8690_split__nat,axiom,
    ! [P2: nat > $o,I2: int] :
      ( ( P2 @ ( nat2 @ I2 ) )
      = ( ! [N4: nat] :
            ( ( I2
              = ( semiri1314217659103216013at_int @ N4 ) )
           => ( P2 @ N4 ) )
        & ( ( ord_less_int @ I2 @ zero_zero_int )
         => ( P2 @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_8691_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_8692_nat__mult__distrib,axiom,
    ! [Z3: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_8693_Suc__as__int,axiom,
    ( suc
    = ( ^ [A5: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A5 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_8694_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_8695_nat__floor__neg,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ X @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_8696_div__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_8697_floor__eq3,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq3
thf(fact_8698_le__nat__floor,axiom,
    ! [X: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X ) @ A )
     => ( ord_less_eq_nat @ X @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_8699_mod__abs__eq__div__nat,axiom,
    ! [K2: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K2 ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K2 ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_8700_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_8701_sgn__power__injE,axiom,
    ! [A: real,N: nat,X: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N ) )
        = X )
     => ( ( X
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_8702_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
     => ( ( suc @ ( nat2 @ Z3 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z3 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_8703_nat__mult__distrib__neg,axiom,
    ! [Z3: int,Z6: int] :
      ( ( ord_less_eq_int @ Z3 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z3 @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_8704_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_8705_floor__eq4,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N ) @ X )
     => ( ( ord_less_real @ X @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X ) )
          = N ) ) ) ).

% floor_eq4
thf(fact_8706_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_8707_diff__nat__eq__if,axiom,
    ! [Z6: int,Z3: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z3 ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z3 @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z3 @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_8708_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_8709_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_8710_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_8711_nat__dvd__iff,axiom,
    ! [Z3: int,M2: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z3 ) @ M2 )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( dvd_dvd_int @ Z3 @ ( semiri1314217659103216013at_int @ M2 ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z3 )
         => ( M2 = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_8712_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_8713_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_8714_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% signed_take_bit_eq_take_bit_shift
thf(fact_8715_arctan__inverse,axiom,
    ! [X: real] :
      ( ( X != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X ) ) ) ) ).

% arctan_inverse
thf(fact_8716_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_8717_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_8718_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_8719_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_8720_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_8721_real__sqrt__mult__self,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sqrt @ A ) @ ( sqrt @ A ) )
      = ( abs_abs_real @ A ) ) ).

% real_sqrt_mult_self
thf(fact_8722_real__sqrt__abs2,axiom,
    ! [X: real] :
      ( ( sqrt @ ( times_times_real @ X @ X ) )
      = ( abs_abs_real @ X ) ) ).

% real_sqrt_abs2
thf(fact_8723_pred__numeral__inc,axiom,
    ! [K2: num] :
      ( ( pred_numeral @ ( inc @ K2 ) )
      = ( numeral_numeral_nat @ K2 ) ) ).

% pred_numeral_inc
thf(fact_8724_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_8725_real__sqrt__mult,axiom,
    ! [X: real,Y: real] :
      ( ( sqrt @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_mult
thf(fact_8726_num__induct,axiom,
    ! [P2: num > $o,X: num] :
      ( ( P2 @ one )
     => ( ! [X3: num] :
            ( ( P2 @ X3 )
           => ( P2 @ ( inc @ X3 ) ) )
       => ( P2 @ X ) ) ) ).

% num_induct
thf(fact_8727_add__inc,axiom,
    ! [X: num,Y: num] :
      ( ( plus_plus_num @ X @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X @ Y ) ) ) ).

% add_inc
thf(fact_8728_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_8729_inc_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit0 @ X ) )
      = ( bit1 @ X ) ) ).

% inc.simps(2)
thf(fact_8730_inc_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( inc @ ( bit1 @ X ) )
      = ( bit0 @ ( inc @ X ) ) ) ).

% inc.simps(3)
thf(fact_8731_add__One,axiom,
    ! [X: num] :
      ( ( plus_plus_num @ X @ one )
      = ( inc @ X ) ) ).

% add_One
thf(fact_8732_le__real__sqrt__sumsq,axiom,
    ! [X: real,Y: real] : ( ord_less_eq_real @ X @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X @ X ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_8733_inc__BitM__eq,axiom,
    ! [N: num] :
      ( ( inc @ ( bitM @ N ) )
      = ( bit0 @ N ) ) ).

% inc_BitM_eq
thf(fact_8734_BitM__inc__eq,axiom,
    ! [N: num] :
      ( ( bitM @ ( inc @ N ) )
      = ( bit1 @ N ) ) ).

% BitM_inc_eq
thf(fact_8735_mult__inc,axiom,
    ! [X: num,Y: num] :
      ( ( times_times_num @ X @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X @ Y ) @ X ) ) ).

% mult_inc
thf(fact_8736_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X: real,Y: real,Xa2: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_8737_arith__geo__mean__sqrt,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_8738_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_8739_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N4 ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N4 ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_8740_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_8741_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_8742_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_8743_or__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(4)
thf(fact_8744_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_8745_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_8746_or__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% or_nat_numerals(3)
thf(fact_8747_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_8748_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W2: num,N: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_8749_cis__mult,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( cis @ A ) @ ( cis @ B ) )
      = ( cis @ ( plus_plus_real @ A @ B ) ) ) ).

% cis_mult
thf(fact_8750_DeMoivre,axiom,
    ! [A: real,N: nat] :
      ( ( power_power_complex @ ( cis @ A ) @ N )
      = ( cis @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre
thf(fact_8751_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_8752_int__bit__bound,axiom,
    ! [K2: int] :
      ~ ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ N3 @ M3 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ M3 )
                = ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) )
         => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( bit_se1146084159140164899it_int @ K2 @ ( minus_minus_nat @ N3 @ one_one_nat ) )
                = ( ~ ( bit_se1146084159140164899it_int @ K2 @ N3 ) ) ) ) ) ).

% int_bit_bound
thf(fact_8753_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N4: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N4 ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% set_bit_eq
thf(fact_8754_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N4: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N4 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ) ).

% unset_bit_eq
thf(fact_8755_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_8756_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_8757_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_8758_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_8759_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_8760_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_8761_not__bit__Suc__0__Suc,axiom,
    ! [N: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N ) ) ).

% not_bit_Suc_0_Suc
thf(fact_8762_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_8763_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_8764_sin__times__pi__eq__0,axiom,
    ! [X: real] :
      ( ( ( sin_real @ ( times_times_real @ X @ pi ) )
        = zero_zero_real )
      = ( member_real @ X @ ring_1_Ints_real ) ) ).

% sin_times_pi_eq_0
thf(fact_8765_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_8766_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_8767_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_8768_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_8769_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_8770_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_8771_complex__i__mult__minus,axiom,
    ! [X: complex] :
      ( ( times_times_complex @ imaginary_unit @ ( times_times_complex @ imaginary_unit @ X ) )
      = ( uminus1482373934393186551omplex @ X ) ) ).

% complex_i_mult_minus
thf(fact_8772_divide__numeral__i,axiom,
    ! [Z3: complex,N: num] :
      ( ( divide1717551699836669952omplex @ Z3 @ ( times_times_complex @ ( numera6690914467698888265omplex @ N ) @ imaginary_unit ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z3 ) ) @ ( numera6690914467698888265omplex @ N ) ) ) ).

% divide_numeral_i
thf(fact_8773_divide__i,axiom,
    ! [X: complex] :
      ( ( divide1717551699836669952omplex @ X @ imaginary_unit )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ imaginary_unit ) @ X ) ) ).

% divide_i
thf(fact_8774_i__squared,axiom,
    ( ( times_times_complex @ imaginary_unit @ imaginary_unit )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% i_squared
thf(fact_8775_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_8776_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_8777_xor__nat__numerals_I3_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).

% xor_nat_numerals(3)
thf(fact_8778_xor__nat__numerals_I4_J,axiom,
    ! [X: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).

% xor_nat_numerals(4)
thf(fact_8779_i__times__eq__iff,axiom,
    ! [W2: complex,Z3: complex] :
      ( ( ( times_times_complex @ imaginary_unit @ W2 )
        = Z3 )
      = ( W2
        = ( uminus1482373934393186551omplex @ ( times_times_complex @ imaginary_unit @ Z3 ) ) ) ) ).

% i_times_eq_iff
thf(fact_8780_i__mult__Complex,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ imaginary_unit @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% i_mult_Complex
thf(fact_8781_Complex__mult__i,axiom,
    ! [A: real,B: real] :
      ( ( times_times_complex @ ( complex2 @ A @ B ) @ imaginary_unit )
      = ( complex2 @ ( uminus_uminus_real @ B ) @ A ) ) ).

% Complex_mult_i
thf(fact_8782_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N4: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N4 @ ( if_nat @ ( N4 = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_8783_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N4: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_8784_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_8785_Least__eq__0,axiom,
    ! [P2: nat > $o] :
      ( ( P2 @ zero_zero_nat )
     => ( ( ord_Least_nat @ P2 )
        = zero_zero_nat ) ) ).

% Least_eq_0
thf(fact_8786_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_8787_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_8788_atLeastLessThan__singleton,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ ( suc @ M2 ) )
      = ( insert_nat @ M2 @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_8789_all__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
           => ( P2 @ M6 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_8790_ex__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N )
            & ( P2 @ M6 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_8791_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_8792_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_8793_atLeastLessThan0,axiom,
    ! [M2: nat] :
      ( ( set_or4665077453230672383an_nat @ M2 @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_8794_Least__Suc2,axiom,
    ! [P2: nat > $o,N: nat,Q: nat > $o,M2: nat] :
      ( ( P2 @ N )
     => ( ( Q @ M2 )
       => ( ~ ( P2 @ zero_zero_nat )
         => ( ! [K: nat] :
                ( ( P2 @ ( suc @ K ) )
                = ( Q @ K ) )
           => ( ( ord_Least_nat @ P2 )
              = ( suc @ ( ord_Least_nat @ Q ) ) ) ) ) ) ) ).

% Least_Suc2
thf(fact_8795_Least__Suc,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ( ( ord_Least_nat @ P2 )
          = ( suc
            @ ( ord_Least_nat
              @ ^ [M6: nat] : ( P2 @ ( suc @ M6 ) ) ) ) ) ) ) ).

% Least_Suc
thf(fact_8796_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_8797_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N6: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( finite_finite_nat @ N6 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_8798_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_8799_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J2 @ K2 ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) @ ( set_or4665077453230672383an_nat @ J2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_8800_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_8801_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_8802_prod__Suc__fact,axiom,
    ! [N: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
      = ( semiri1408675320244567234ct_nat @ N ) ) ).

% prod_Suc_fact
thf(fact_8803_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N6: set_nat,N: nat] :
      ( ( ord_less_eq_set_nat @ N6 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N6 ) @ N ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_8804_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_8805_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_8806_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_8807_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_8808_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_8809_Chebyshev__sum__upper__nat,axiom,
    ! [N: nat,A: nat > nat,B: nat > nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_eq_nat @ I3 @ J3 )
         => ( ( ord_less_nat @ J3 @ N )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J3 ) ) ) )
     => ( ! [I3: nat,J3: nat] :
            ( ( ord_less_eq_nat @ I3 @ J3 )
           => ( ( ord_less_nat @ J3 @ N )
             => ( ord_less_eq_nat @ ( B @ J3 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N
            @ ( groups3542108847815614940at_nat
              @ ^ [I: nat] : ( times_times_nat @ ( A @ I ) @ ( B @ I ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_8810_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_8811_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_8812_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_8813_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_atLeastLessThan_int
thf(fact_8814_exp__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ pi ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i'
thf(fact_8815_exp__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ imaginary_unit ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% exp_pi_i
thf(fact_8816_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_8817_complex__exp__exists,axiom,
    ! [Z3: complex] :
    ? [A3: complex,R4: real] :
      ( Z3
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( exp_complex @ A3 ) ) ) ).

% complex_exp_exists
thf(fact_8818_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_8819_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_8820_Complex__mult__complex__of__real,axiom,
    ! [X: real,Y: real,R3: real] :
      ( ( times_times_complex @ ( complex2 @ X @ Y ) @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ ( times_times_real @ X @ R3 ) @ ( times_times_real @ Y @ R3 ) ) ) ).

% Complex_mult_complex_of_real
thf(fact_8821_complex__of__real__mult__Complex,axiom,
    ! [R3: real,X: real,Y: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( complex2 @ X @ Y ) )
      = ( complex2 @ ( times_times_real @ R3 @ X ) @ ( times_times_real @ R3 @ Y ) ) ) ).

% complex_of_real_mult_Complex
thf(fact_8822_Complex__eq,axiom,
    ( complex2
    = ( ^ [A5: real,B4: real] : ( plus_plus_complex @ ( real_V4546457046886955230omplex @ A5 ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% Complex_eq
thf(fact_8823_cis__conv__exp,axiom,
    ( cis
    = ( ^ [B4: real] : ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ B4 ) ) ) ) ) ).

% cis_conv_exp
thf(fact_8824_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_8825_complex__of__real__i,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ imaginary_unit )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% complex_of_real_i
thf(fact_8826_i__complex__of__real,axiom,
    ! [R3: real] :
      ( ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ R3 ) )
      = ( complex2 @ zero_zero_real @ R3 ) ) ).

% i_complex_of_real
thf(fact_8827_complex__split__polar,axiom,
    ! [Z3: complex] :
    ? [R4: real,A3: real] :
      ( Z3
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ R4 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A3 ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A3 ) ) ) ) ) ) ).

% complex_split_polar
thf(fact_8828_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_8829_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_8830_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_8831_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_8832_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_8833_cmod__complex__polar,axiom,
    ! [R3: real,A: real] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ R3 ) @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) ) )
      = ( abs_abs_real @ R3 ) ) ).

% cmod_complex_polar
thf(fact_8834_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_8835_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_8836_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_8837_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_8838_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_8839_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X7: nat > real] :
        ! [J: nat] :
        ? [M9: nat] :
        ! [M6: nat] :
          ( ( ord_less_eq_nat @ M9 @ M6 )
         => ! [N4: nat] :
              ( ( ord_less_eq_nat @ M9 @ N4 )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_8840_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_8841_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_8842_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_8843_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A5: complex,B4: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A5 @ ( cnj @ B4 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_8844_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,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
          @ ( nat_prod_decode @ N ) ) ) ) ).

% list_decode.psimps(2)
thf(fact_8845_complex__cnj__mult,axiom,
    ! [X: complex,Y: complex] :
      ( ( cnj @ ( times_times_complex @ X @ Y ) )
      = ( times_times_complex @ ( cnj @ X ) @ ( cnj @ Y ) ) ) ).

% complex_cnj_mult
thf(fact_8846_list__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_list_decode @ X )
        = ( nat_list_decode @ Y ) )
      = ( X = Y ) ) ).

% list_decode_eq
thf(fact_8847_times__integer__code_I1_J,axiom,
    ! [K2: code_integer] :
      ( ( times_3573771949741848930nteger @ K2 @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(1)
thf(fact_8848_times__integer__code_I2_J,axiom,
    ! [L: code_integer] :
      ( ( times_3573771949741848930nteger @ zero_z3403309356797280102nteger @ L )
      = zero_z3403309356797280102nteger ) ).

% times_integer_code(2)
thf(fact_8849_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_8850_list__decode_Osimps_I2_J,axiom,
    ! [N: nat] :
      ( ( nat_list_decode @ ( suc @ N ) )
      = ( produc2761476792215241774st_nat
        @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
        @ ( nat_prod_decode @ N ) ) ) ).

% list_decode.simps(2)
thf(fact_8851_complex__mod__mult__cnj,axiom,
    ! [Z3: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_8852_complex__norm__square,axiom,
    ! [Z3: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) ) ).

% complex_norm_square
thf(fact_8853_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_8854_list__decode_Opelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( accp_nat @ nat_list_decode_rel @ X )
       => ( ( ( X = zero_zero_nat )
           => ( ( Y = nil_nat )
             => ~ ( accp_nat @ nat_list_decode_rel @ zero_zero_nat ) ) )
         => ~ ! [N3: nat] :
                ( ( X
                  = ( suc @ N3 ) )
               => ( ( Y
                    = ( produc2761476792215241774st_nat
                      @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
                      @ ( nat_prod_decode @ N3 ) ) )
                 => ~ ( accp_nat @ nat_list_decode_rel @ ( suc @ N3 ) ) ) ) ) ) ) ).

% list_decode.pelims
thf(fact_8855_list__decode_Oelims,axiom,
    ! [X: nat,Y: list_nat] :
      ( ( ( nat_list_decode @ X )
        = Y )
     => ( ( ( X = zero_zero_nat )
         => ( Y != nil_nat ) )
       => ~ ! [N3: nat] :
              ( ( X
                = ( suc @ N3 ) )
             => ( Y
               != ( produc2761476792215241774st_nat
                  @ ^ [X4: nat,Y4: nat] : ( cons_nat @ X4 @ ( nat_list_decode @ Y4 ) )
                  @ ( nat_prod_decode @ N3 ) ) ) ) ) ) ).

% list_decode.elims
thf(fact_8856_cnj__add__mult__eq__Re,axiom,
    ! [Z3: complex,W2: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z3 @ ( cnj @ W2 ) ) @ ( times_times_complex @ ( cnj @ Z3 ) @ W2 ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z3 @ ( cnj @ W2 ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_8857_list__encode_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ~ ! [X3: nat,Xs3: list_nat] :
            ( X
           != ( cons_nat @ X3 @ Xs3 ) ) ) ).

% list_encode.cases
thf(fact_8858_times__integer_Oabs__eq,axiom,
    ! [Xa2: int,X: int] :
      ( ( times_3573771949741848930nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X ) )
      = ( code_integer_of_int @ ( times_times_int @ Xa2 @ X ) ) ) ).

% times_integer.abs_eq
thf(fact_8859_scaleR__complex_Osimps_I1_J,axiom,
    ! [R3: real,X: complex] :
      ( ( re @ ( real_V2046097035970521341omplex @ R3 @ X ) )
      = ( times_times_real @ R3 @ ( re @ X ) ) ) ).

% scaleR_complex.simps(1)
thf(fact_8860_list__decode_Osimps_I1_J,axiom,
    ( ( nat_list_decode @ zero_zero_nat )
    = nil_nat ) ).

% list_decode.simps(1)
thf(fact_8861_Re__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( re @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Re_complex_div_eq_0
thf(fact_8862_complex__mod__sqrt__Re__mult__cnj,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z4: complex] : ( sqrt @ ( re @ ( times_times_complex @ Z4 @ ( cnj @ Z4 ) ) ) ) ) ) ).

% complex_mod_sqrt_Re_mult_cnj
thf(fact_8863_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_8864_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_8865_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_8866_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_8867_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_8868_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_8869_complex__add__cnj,axiom,
    ! [Z3: complex] :
      ( ( plus_plus_complex @ Z3 @ ( cnj @ Z3 ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z3 ) ) ) ) ).

% complex_add_cnj
thf(fact_8870_list__encode_Oelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( ( X = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X3: nat,Xs3: list_nat] :
              ( ( X
                = ( cons_nat @ X3 @ Xs3 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_8871_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X4: complex,Y4: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_8872_Re__Reals__divide,axiom,
    ! [R3: complex,Z3: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R3 @ Z3 ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R3 ) @ ( re @ Z3 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_8873_list__encode__inverse,axiom,
    ! [X: list_nat] :
      ( ( nat_list_decode @ ( nat_list_encode @ X ) )
      = X ) ).

% list_encode_inverse
thf(fact_8874_list__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_list_encode @ ( nat_list_decode @ N ) )
      = N ) ).

% list_decode_inverse
thf(fact_8875_complex__In__mult__cnj__zero,axiom,
    ! [Z3: complex] :
      ( ( im @ ( times_times_complex @ Z3 @ ( cnj @ Z3 ) ) )
      = zero_zero_real ) ).

% complex_In_mult_cnj_zero
thf(fact_8876_Im__i__times,axiom,
    ! [Z3: complex] :
      ( ( im @ ( times_times_complex @ imaginary_unit @ Z3 ) )
      = ( re @ Z3 ) ) ).

% Im_i_times
thf(fact_8877_real__eq__imaginary__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( X
            = ( times_times_complex @ imaginary_unit @ Y ) )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% real_eq_imaginary_iff
thf(fact_8878_imaginary__eq__real__iff,axiom,
    ! [Y: complex,X: complex] :
      ( ( member_complex @ Y @ real_V2521375963428798218omplex )
     => ( ( member_complex @ X @ real_V2521375963428798218omplex )
       => ( ( ( times_times_complex @ imaginary_unit @ Y )
            = X )
          = ( ( X = zero_zero_complex )
            & ( Y = zero_zero_complex ) ) ) ) ) ).

% imaginary_eq_real_iff
thf(fact_8879_Re__i__times,axiom,
    ! [Z3: complex] :
      ( ( re @ ( times_times_complex @ imaginary_unit @ Z3 ) )
      = ( uminus_uminus_real @ ( im @ Z3 ) ) ) ).

% Re_i_times
thf(fact_8880_list__encode__eq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( nat_list_encode @ X )
        = ( nat_list_encode @ Y ) )
      = ( X = Y ) ) ).

% list_encode_eq
thf(fact_8881_scaleR__complex_Osimps_I2_J,axiom,
    ! [R3: real,X: complex] :
      ( ( im @ ( real_V2046097035970521341omplex @ R3 @ X ) )
      = ( times_times_real @ R3 @ ( im @ X ) ) ) ).

% scaleR_complex.simps(2)
thf(fact_8882_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_8883_times__complex_Osimps_I2_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( times_times_complex @ X @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) ) ) ).

% times_complex.simps(2)
thf(fact_8884_times__complex_Osimps_I1_J,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( times_times_complex @ X @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) ) ).

% times_complex.simps(1)
thf(fact_8885_Im__complex__div__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( im @ ( divide1717551699836669952omplex @ A @ B ) )
        = zero_zero_real )
      = ( ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) )
        = zero_zero_real ) ) ).

% Im_complex_div_eq_0
thf(fact_8886_scaleR__complex_Ocode,axiom,
    ( real_V2046097035970521341omplex
    = ( ^ [R: real,X4: complex] : ( complex2 @ ( times_times_real @ R @ ( re @ X4 ) ) @ ( times_times_real @ R @ ( im @ X4 ) ) ) ) ) ).

% scaleR_complex.code
thf(fact_8887_Im__Reals__divide,axiom,
    ! [R3: complex,Z3: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R3 @ Z3 ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R3 ) ) @ ( im @ Z3 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_8888_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_8889_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_8890_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_8891_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_8892_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_8893_Re__exp,axiom,
    ! [Z3: complex] :
      ( ( re @ ( exp_complex @ Z3 ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z3 ) ) @ ( cos_real @ ( im @ Z3 ) ) ) ) ).

% Re_exp
thf(fact_8894_Im__exp,axiom,
    ! [Z3: complex] :
      ( ( im @ ( exp_complex @ Z3 ) )
      = ( times_times_real @ ( exp_real @ ( re @ Z3 ) ) @ ( sin_real @ ( im @ Z3 ) ) ) ) ).

% Im_exp
thf(fact_8895_complex__eq,axiom,
    ! [A: complex] :
      ( A
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( re @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( im @ A ) ) ) ) ) ).

% complex_eq
thf(fact_8896_times__complex_Ocode,axiom,
    ( times_times_complex
    = ( ^ [X4: complex,Y4: complex] : ( complex2 @ ( minus_minus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y4 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y4 ) ) ) @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y4 ) ) @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y4 ) ) ) ) ) ) ).

% times_complex.code
thf(fact_8897_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_8898_exp__eq__polar,axiom,
    ( exp_complex
    = ( ^ [Z4: complex] : ( times_times_complex @ ( real_V4546457046886955230omplex @ ( exp_real @ ( re @ Z4 ) ) ) @ ( cis @ ( im @ Z4 ) ) ) ) ) ).

% exp_eq_polar
thf(fact_8899_Im__power2,axiom,
    ! [X: complex] :
      ( ( im @ ( power_power_complex @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).

% Im_power2
thf(fact_8900_list__encode_Osimps_I2_J,axiom,
    ! [X: nat,Xs2: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X @ Xs2 ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X @ ( nat_list_encode @ Xs2 ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_8901_complex__diff__cnj,axiom,
    ! [Z3: complex] :
      ( ( minus_minus_complex @ Z3 @ ( cnj @ Z3 ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z3 ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_8902_Re__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_8903_complex__mult__cnj,axiom,
    ! [Z3: complex] :
      ( ( times_times_complex @ Z3 @ ( cnj @ Z3 ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_8904_Im__divide,axiom,
    ! [X: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_8905_complex__abs__le__norm,axiom,
    ! [Z3: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z3 ) ) @ ( abs_abs_real @ ( im @ Z3 ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z3 ) ) ) ).

% complex_abs_le_norm
thf(fact_8906_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_8907_csqrt_Osimps_I2_J,axiom,
    ! [Z3: complex] :
      ( ( im @ ( csqrt @ Z3 ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z3 )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z3 ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_8908_csqrt__of__real__nonpos,axiom,
    ! [X: complex] :
      ( ( ( im @ X )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X ) @ zero_zero_real )
       => ( ( csqrt @ X )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_8909_csqrt__minus,axiom,
    ! [X: complex] :
      ( ( ( ord_less_real @ ( im @ X ) @ zero_zero_real )
        | ( ( ( im @ X )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).

% csqrt_minus
thf(fact_8910_bezw__0,axiom,
    ! [X: nat] :
      ( ( bezw @ X @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_8911_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_8912_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_8913_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_8914_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_8915_bit__push__bit__iff__nat,axiom,
    ! [M2: nat,Q2: nat,N: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M2 @ Q2 ) @ N )
      = ( ( ord_less_eq_nat @ M2 @ N )
        & ( bit_se1148574629649215175it_nat @ Q2 @ ( minus_minus_nat @ N @ M2 ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_8916_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_8917_max__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_max_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ M7 @ N ) )
        @ M2 ) ) ).

% max_Suc2
thf(fact_8918_max__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_max_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ ( suc @ N )
        @ ^ [M7: nat] : ( suc @ ( ord_max_nat @ N @ M7 ) )
        @ M2 ) ) ).

% max_Suc1
thf(fact_8919_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_8920_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N4: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_nat_def
thf(fact_8921_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N4: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N4 ) ) ) ) ).

% push_bit_int_def
thf(fact_8922_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_8923_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_8924_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_8925_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,J: code_integer] : ( if_nat @ ( J = 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_8926_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_8927_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_8928_fst__divmod__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M2 @ N ) )
      = ( divide_divide_nat @ M2 @ N ) ) ).

% fst_divmod_nat
thf(fact_8929_nat__of__integer__non__positive,axiom,
    ! [K2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K2 @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K2 )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_8930_snd__divmod__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M2 @ N ) )
      = ( modulo_modulo_nat @ M2 @ N ) ) ).

% snd_divmod_nat
thf(fact_8931_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_8932_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_8933_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,J: code_integer] : ( if_int @ ( J = 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_8934_times__integer_Orep__eq,axiom,
    ! [X: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( times_3573771949741848930nteger @ X @ Xa2 ) )
      = ( times_times_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% times_integer.rep_eq
thf(fact_8935_bezw__non__0,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_8936_bezw_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_8937_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X4: nat,Y4: nat] : ( if_Pro3027730157355071871nt_int @ ( Y4 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y4 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_8938_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_8939_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_8940_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_8941_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_8942_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R: int] : ( if_int @ ( R = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_8943_bezw_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_8944_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_8945_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_8946_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_8947_prod__decode__aux_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_8948_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_8949_Suc__funpow,axiom,
    ! [N: nat] :
      ( ( compow_nat_nat @ N @ suc )
      = ( plus_plus_nat @ N ) ) ).

% Suc_funpow
thf(fact_8950_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_8951_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_8952_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_8953_bij__betw__nth__root__unity,axiom,
    ! [C2: complex,N: nat] :
      ( ( C2 != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N @ ( real_V1022390504157884413omplex @ C2 ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C2 ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z4: complex] :
                ( ( power_power_complex @ Z4 @ N )
                = C2 ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_8954_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_8955_real__root__Suc__0,axiom,
    ! [X: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X )
      = X ) ).

% real_root_Suc_0
thf(fact_8956_real__root__eq__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = ( root @ N @ Y ) )
        = ( X = Y ) ) ) ).

% real_root_eq_iff
thf(fact_8957_root__0,axiom,
    ! [X: real] :
      ( ( root @ zero_zero_nat @ X )
      = zero_zero_real ) ).

% root_0
thf(fact_8958_real__root__eq__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = zero_zero_real )
        = ( X = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_8959_real__root__less__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_real @ X @ Y ) ) ) ).

% real_root_less_iff
thf(fact_8960_real__root__le__iff,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ X @ Y ) ) ) ).

% real_root_le_iff
thf(fact_8961_real__root__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ one_one_real )
        = one_one_real ) ) ).

% real_root_one
thf(fact_8962_real__root__eq__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( root @ N @ X )
          = one_one_real )
        = ( X = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_8963_real__root__gt__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_8964_real__root__lt__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_real @ X @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_8965_real__root__ge__0__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_8966_real__root__le__0__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ zero_zero_real )
        = ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_8967_real__root__gt__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_8968_real__root__lt__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_real @ X @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_8969_real__root__ge__1__iff,axiom,
    ! [N: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_8970_real__root__le__1__iff,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ ( root @ N @ X ) @ one_one_real )
        = ( ord_less_eq_real @ X @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_8971_real__root__pow__pos2,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos2
thf(fact_8972_real__root__mult__exp,axiom,
    ! [M2: nat,N: nat,X: real] :
      ( ( root @ ( times_times_nat @ M2 @ N ) @ X )
      = ( root @ M2 @ ( root @ N @ X ) ) ) ).

% real_root_mult_exp
thf(fact_8973_real__root__mult,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( root @ N @ ( times_times_real @ X @ Y ) )
      = ( times_times_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ).

% real_root_mult
thf(fact_8974_real__root__less__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ X @ Y )
       => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_8975_real__root__le__mono,axiom,
    ! [N: nat,X: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ X @ Y )
       => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_8976_real__root__power,axiom,
    ! [N: nat,X: real,K2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( power_power_real @ X @ K2 ) )
        = ( power_power_real @ ( root @ N @ X ) @ K2 ) ) ) ).

% real_root_power
thf(fact_8977_real__root__abs,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( root @ N @ ( abs_abs_real @ X ) )
        = ( abs_abs_real @ ( root @ N @ X ) ) ) ) ).

% real_root_abs
thf(fact_8978_sgn__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( sgn_sgn_real @ ( root @ N @ X ) )
        = ( sgn_sgn_real @ X ) ) ) ).

% sgn_root
thf(fact_8979_real__root__gt__zero,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_gt_zero
thf(fact_8980_real__root__strict__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ( ord_less_real @ one_one_real @ X )
         => ( ord_less_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_8981_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_8982_real__root__pos__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N @ X ) ) ) ) ).

% real_root_pos_pos
thf(fact_8983_real__root__strict__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ N @ N6 )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( ord_less_real @ X @ one_one_real )
           => ( ord_less_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_8984_real__root__decreasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N6 )
       => ( ( ord_less_eq_real @ one_one_real @ X )
         => ( ord_less_eq_real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).

% real_root_decreasing
thf(fact_8985_real__root__pow__pos,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( power_power_real @ ( root @ N @ X ) @ N )
          = X ) ) ) ).

% real_root_pow_pos
thf(fact_8986_real__root__pos__unique,axiom,
    ! [N: nat,Y: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N )
            = X )
         => ( ( root @ N @ X )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_8987_real__root__power__cancel,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_real @ zero_zero_real @ X )
       => ( ( root @ N @ ( power_power_real @ X @ N ) )
          = X ) ) ) ).

% real_root_power_cancel
thf(fact_8988_real__root__increasing,axiom,
    ! [N: nat,N6: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_eq_nat @ N @ N6 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X )
         => ( ( ord_less_eq_real @ X @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_8989_sgn__power__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N @ X ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N @ X ) ) @ N ) )
        = X ) ) ).

% sgn_power_root
thf(fact_8990_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_8991_ln__root,axiom,
    ! [N: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% ln_root
thf(fact_8992_log__root,axiom,
    ! [N: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log2 @ B @ ( root @ N @ A ) )
          = ( divide_divide_real @ ( log2 @ B @ A ) @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ).

% log_root
thf(fact_8993_log__base__root,axiom,
    ! [N: nat,B: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log2 @ ( root @ N @ B ) @ X )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( log2 @ B @ X ) ) ) ) ) ).

% log_base_root
thf(fact_8994_split__root,axiom,
    ! [P2: real > $o,N: nat,X: real] :
      ( ( P2 @ ( root @ N @ X ) )
      = ( ( ( N = zero_zero_nat )
         => ( P2 @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N )
         => ! [Y4: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) )
                = X )
             => ( P2 @ Y4 ) ) ) ) ) ).

% split_root
thf(fact_8995_root__powr__inverse,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( ( root @ N @ X )
          = ( powr_real @ X @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_8996_powr__powr,axiom,
    ! [X: real,A: real,B: real] :
      ( ( powr_real @ ( powr_real @ X @ A ) @ B )
      = ( powr_real @ X @ ( times_times_real @ A @ B ) ) ) ).

% powr_powr
thf(fact_8997_powr__mult,axiom,
    ! [X: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_8998_divide__powr__uminus,axiom,
    ! [A: real,B: real,C2: real] :
      ( ( divide_divide_real @ A @ ( powr_real @ B @ C2 ) )
      = ( times_times_real @ A @ ( powr_real @ B @ ( uminus_uminus_real @ C2 ) ) ) ) ).

% divide_powr_uminus
thf(fact_8999_ln__powr,axiom,
    ! [X: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( ln_ln_real @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( ln_ln_real @ X ) ) ) ) ).

% ln_powr
thf(fact_9000_log__powr,axiom,
    ! [X: real,B: real,Y: real] :
      ( ( X != zero_zero_real )
     => ( ( log2 @ B @ ( powr_real @ X @ Y ) )
        = ( times_times_real @ Y @ ( log2 @ B @ X ) ) ) ) ).

% log_powr
thf(fact_9001_powr__mult__base,axiom,
    ! [X: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X )
     => ( ( times_times_real @ X @ ( powr_real @ X @ Y ) )
        = ( powr_real @ X @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_9002_ln__powr__bound2,axiom,
    ! [X: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X ) ) ) ) ).

% ln_powr_bound2
thf(fact_9003_log__add__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ ( log2 @ B @ X ) @ Y )
            = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_9004_add__log__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( plus_plus_real @ Y @ ( log2 @ B @ X ) )
            = ( log2 @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_9005_log__minus__eq__powr,axiom,
    ! [B: real,X: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X )
         => ( ( minus_minus_real @ ( log2 @ B @ X ) @ Y )
            = ( log2 @ B @ ( times_times_real @ X @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_9006_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) )
          @ Xa2
          @ X ) ) ) ).

% times_int.abs_eq
thf(fact_9007_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_9008_num__of__nat__numeral__eq,axiom,
    ! [Q2: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q2 ) )
      = Q2 ) ).

% num_of_nat_numeral_eq
thf(fact_9009_eq__Abs__Integ,axiom,
    ! [Z3: int] :
      ~ ! [X3: nat,Y3: nat] :
          ( Z3
         != ( abs_Integ @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) ) ).

% eq_Abs_Integ
thf(fact_9010_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9011_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9012_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N4: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N4 @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9013_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_9014_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_9015_uminus__int_Oabs__eq,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 )
          @ X ) ) ) ).

% uminus_int.abs_eq
thf(fact_9016_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9017_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_9018_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_9019_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) )
        @ Xa2
        @ X ) ) ).

% less_eq_int.abs_eq
thf(fact_9020_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V3 ) ) )
          @ Xa2
          @ X ) ) ) ).

% plus_int.abs_eq
thf(fact_9021_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X4: nat,Y4: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) )
          @ Xa2
          @ X ) ) ) ).

% minus_int.abs_eq
thf(fact_9022_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_9023_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_9024_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9025_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9026_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X4: int,Xa3: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y4: nat,Z4: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y4 @ V3 ) @ ( plus_plus_nat @ U2 @ Z4 ) ) )
          @ ( rep_Integ @ X4 )
          @ ( rep_Integ @ Xa3 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9027_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C2: nat,Y: nat,X: nat] :
      ( ( ( ord_less_nat @ C2 @ Y )
       => ( ( image_nat_nat
            @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
            @ ( set_or4665077453230672383an_nat @ X @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X @ C2 ) @ ( minus_minus_nat @ Y @ C2 ) ) ) )
      & ( ~ ( ord_less_nat @ C2 @ Y )
       => ( ( ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X @ Y )
           => ( ( image_nat_nat
                @ ^ [I: nat] : ( minus_minus_nat @ I @ C2 )
                @ ( set_or4665077453230672383an_nat @ X @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9028_bij__betw__Suc,axiom,
    ! [M5: set_nat,N6: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M5 @ N6 )
      = ( ( image_nat_nat @ suc @ M5 )
        = N6 ) ) ).

% bij_betw_Suc
thf(fact_9029_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J2 ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9030_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9031_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_9032_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_9033_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_9034_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_9035_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_9036_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_9037_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_9038_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9039_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) ) ) ) ).

% uminus_int_def
thf(fact_9040_rat__inverse__code,axiom,
    ! [P: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,B4: int] : ( if_Pro3027730157355071871nt_int @ ( A5 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A5 ) @ B4 ) @ ( abs_abs_int @ A5 ) ) )
        @ ( quotient_of @ P ) ) ) ).

% rat_inverse_code
thf(fact_9041_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9042_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_9043_divide__rat__def,axiom,
    ( divide_divide_rat
    = ( ^ [Q5: rat,R: rat] : ( times_times_rat @ Q5 @ ( inverse_inverse_rat @ R ) ) ) ) ).

% divide_rat_def
thf(fact_9044_rat__less__eq__code,axiom,
    ( ord_less_eq_rat
    = ( ^ [P5: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_eq_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_eq_code
thf(fact_9045_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P5: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A5: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B4: int,D2: int] : ( ord_less_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P5 ) ) ) ) ).

% rat_less_code
thf(fact_9046_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9047_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9048_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9049_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9050_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X4: int] : ( plus_plus_int @ X4 @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9051_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9052_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9053_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9054_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X4: nat,Y4: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V3 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9055_rat__minus__code,axiom,
    ! [P: rat,Q2: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P ) ) ) ).

% rat_minus_code
thf(fact_9056_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9057_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_9058_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9059_normalize__crossproduct,axiom,
    ! [Q2: int,S2: int,P: int,R3: int] :
      ( ( Q2 != zero_zero_int )
     => ( ( S2 != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P @ Q2 ) )
            = ( normalize @ ( product_Pair_int_int @ R3 @ S2 ) ) )
         => ( ( times_times_int @ P @ S2 )
            = ( times_times_int @ R3 @ Q2 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9060_rat__times__code,axiom,
    ! [P: rat,Q2: rat] :
      ( ( quotient_of @ ( times_times_rat @ P @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ B4 ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P ) ) ) ).

% rat_times_code
thf(fact_9061_rat__divide__code,axiom,
    ! [P: rat,Q2: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ C3 @ B4 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P ) ) ) ).

% rat_divide_code
thf(fact_9062_rat__plus__code,axiom,
    ! [P: rat,Q2: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P @ Q2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A5: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B4: int,D2: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A5 @ D2 ) @ ( times_times_int @ B4 @ C3 ) ) @ ( times_times_int @ C3 @ D2 ) ) )
            @ ( quotient_of @ Q2 ) )
        @ ( quotient_of @ P ) ) ) ).

% rat_plus_code
thf(fact_9063_list__encode_Opelims,axiom,
    ! [X: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X )
       => ( ( ( X = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X3: nat,Xs3: list_nat] :
                ( ( X
                  = ( cons_nat @ X3 @ Xs3 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9064_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_9065_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_9066_length__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J2 ) )
      = ( minus_minus_nat @ J2 @ I2 ) ) ).

% length_upt
thf(fact_9067_take__upt,axiom,
    ! [I2: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M2 ) @ N )
     => ( ( take_nat @ M2 @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M2 ) ) ) ) ).

% take_upt
thf(fact_9068_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9069_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( upt @ I2 @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_9070_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N4: nat,M6: nat] : ( set_nat2 @ ( upt @ N4 @ ( suc @ M6 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9071_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).

% atLeast_upt
thf(fact_9072_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9073_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_9074_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N4: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ ( suc @ M6 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_9075_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N4: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N4 ) @ M6 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9076_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N4 ) ) ) ) ) ).

% atMost_upto
thf(fact_9077_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_9078_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J2 @ K2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K2 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_9079_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9080_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_9081_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_9082_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9083_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_9084_mono__ge2__power__minus__self,axiom,
    ! [K2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K2 )
     => ( order_mono_nat_nat
        @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K2 @ M6 ) @ M6 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9085_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_9086_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_9087_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_9088_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_9089_card__length__sum__list__rec,axiom,
    ! [M2: nat,N6: 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 )
                  = N6 ) ) ) )
        = ( 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 )
                    = N6 ) ) ) )
          @ ( 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 )
                    = N6 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9090_card__length__sum__list,axiom,
    ! [M2: nat,N6: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M2 )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N6 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N6 @ M2 ) @ one_one_nat ) @ N6 ) ) ).

% card_length_sum_list
thf(fact_9091_range__mod,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( image_nat_nat
          @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) ) ).

% range_mod
thf(fact_9092_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_9093_range__mult,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
      & ( ( A != zero_zero_real )
       => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
          = top_top_set_real ) ) ) ).

% range_mult
thf(fact_9094_sorted__upt,axiom,
    ! [M2: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M2 @ N ) ) ).

% sorted_upt
thf(fact_9095_surj__prod__decode,axiom,
    ( ( image_5846123807819985514at_nat @ nat_prod_decode @ top_top_set_nat )
    = top_to4669805908274784177at_nat ) ).

% surj_prod_decode
thf(fact_9096_bij__prod__decode,axiom,
    bij_be8693218025023041337at_nat @ nat_prod_decode @ top_top_set_nat @ top_to4669805908274784177at_nat ).

% bij_prod_decode
thf(fact_9097_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_9098_bij__list__decode,axiom,
    bij_be6293887246118711976st_nat @ nat_list_decode @ top_top_set_nat @ top_top_set_list_nat ).

% bij_list_decode
thf(fact_9099_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_9100_bij__list__encode,axiom,
    bij_be8532844293280997160at_nat @ nat_list_encode @ top_top_set_list_nat @ top_top_set_nat ).

% bij_list_encode
thf(fact_9101_surj__prod__encode,axiom,
    ( ( image_2486076414777270412at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat )
    = top_top_set_nat ) ).

% surj_prod_encode
thf(fact_9102_bij__prod__encode,axiom,
    bij_be5333170631980326235at_nat @ nat_prod_encode @ top_to4669805908274784177at_nat @ top_top_set_nat ).

% bij_prod_encode
thf(fact_9103_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_9104_root__def,axiom,
    ( root
    = ( ^ [N4: nat,X4: real] :
          ( if_real @ ( N4 = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N4 ) )
            @ X4 ) ) ) ) ).

% root_def
thf(fact_9105_DERIV__even__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
       => ( ( ord_less_real @ X @ zero_zero_real )
         => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N ) ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9106_DERIV__real__root__generic,axiom,
    ! [N: nat,X: real,D3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( X != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
           => ( ( ord_less_real @ zero_zero_real @ X )
             => ( D3
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
             => ( ( ord_less_real @ X @ zero_zero_real )
               => ( D3
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
               => ( D3
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N ) @ D3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9107_DERIV__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9108_DERIV__fun__pow,axiom,
    ! [G2: real > real,M2: real,X: real,N: nat] :
      ( ( has_fi5821293074295781190e_real @ G2 @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X4: real] : ( power_power_real @ ( G2 @ X4 ) @ N )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( G2 @ X ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) @ M2 )
        @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9109_has__real__derivative__powr,axiom,
    ! [Z3: real,R3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z3 )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z4: real] : ( powr_real @ Z4 @ R3 )
        @ ( times_times_real @ R3 @ ( powr_real @ Z3 @ ( minus_minus_real @ R3 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9110_DERIV__series_H,axiom,
    ! [F2: real > nat > real,F5: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N3: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( F2 @ X4 @ N3 )
          @ ( F5 @ X0 @ N3 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X3: real] :
            ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F2 @ X3 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F5 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N3: nat,X3: real,Y3: real] :
                    ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F2 @ X3 @ N3 ) @ ( F2 @ Y3 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X4: real] : ( suminf_real @ ( F2 @ X4 ) )
                  @ ( suminf_real @ ( F5 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9111_DERIV__log,axiom,
    ! [X: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X )
     => ( has_fi5821293074295781190e_real @ ( log2 @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9112_DERIV__fun__powr,axiom,
    ! [G2: real > real,M2: real,X: real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G2 @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G2 @ X ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X4: real] : ( powr_real @ ( G2 @ X4 ) @ R3 )
          @ ( times_times_real @ ( times_times_real @ R3 @ ( powr_real @ ( G2 @ X ) @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M2 )
          @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9113_DERIV__powr,axiom,
    ! [G2: real > real,M2: real,X: real,F2: real > real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G2 @ M2 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G2 @ X ) )
       => ( ( has_fi5821293074295781190e_real @ F2 @ R3 @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] : ( powr_real @ ( G2 @ X4 ) @ ( F2 @ X4 ) )
            @ ( times_times_real @ ( powr_real @ ( G2 @ X ) @ ( F2 @ X ) ) @ ( plus_plus_real @ ( times_times_real @ R3 @ ( ln_ln_real @ ( G2 @ X ) ) ) @ ( divide_divide_real @ ( times_times_real @ M2 @ ( F2 @ X ) ) @ ( G2 @ X ) ) ) )
            @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9114_DERIV__power__series_H,axiom,
    ! [R2: real,F2: nat > real,X0: real] :
      ( ! [X3: real] :
          ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F2 @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X3 @ N4 ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R2 ) @ R2 ) )
       => ( ( ord_less_real @ zero_zero_real @ R2 )
         => ( has_fi5821293074295781190e_real
            @ ^ [X4: real] :
                ( suminf_real
                @ ^ [N4: nat] : ( times_times_real @ ( F2 @ N4 ) @ ( power_power_real @ X4 @ ( suc @ N4 ) ) ) )
            @ ( suminf_real
              @ ^ [N4: nat] : ( times_times_real @ ( times_times_real @ ( F2 @ N4 ) @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) @ ( power_power_real @ X0 @ N4 ) ) )
            @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).

% DERIV_power_series'
thf(fact_9115_DERIV__real__root,axiom,
    ! [N: nat,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_real @ zero_zero_real @ X )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9116_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ! [M: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F2 @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9117_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F2 )
        & ! [M: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
          & ( ( F2 @ X )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                @ ( set_ord_lessThan_nat @ N ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9118_DERIV__odd__real__root,axiom,
    ! [N: nat,X: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
     => ( ( X != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ ( root @ N @ X ) @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9119_Maclaurin__minus,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
      ( ( ord_less_real @ H @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F2 )
         => ( ! [M: nat,T6: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ H @ T6 )
                  & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ H @ T6 )
                & ( ord_less_real @ T6 @ zero_zero_real )
                & ( ( F2 @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9120_Maclaurin2,axiom,
    ! [H: real,Diff: nat > real > real,F2: real > real,N: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M: nat,T6: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ zero_zero_real @ T6 )
                & ( ord_less_eq_real @ T6 @ H ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ? [T6: real] :
              ( ( ord_less_real @ zero_zero_real @ T6 )
              & ( ord_less_eq_real @ T6 @ H )
              & ( ( F2 @ H )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                    @ ( set_ord_lessThan_nat @ N ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9121_Maclaurin,axiom,
    ! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( ( Diff @ zero_zero_nat )
            = F2 )
         => ( ! [M: nat,T6: real] :
                ( ( ( ord_less_nat @ M @ N )
                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
                  & ( ord_less_eq_real @ T6 @ H ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ T6 )
                & ( ord_less_real @ T6 @ H )
                & ( ( F2 @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ H @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9122_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ( X != zero_zero_real )
         => ( ! [M: nat,X3: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
                & ( ( F2 @ X )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9123_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F2 )
     => ( ! [M: nat,T6: real] :
            ( ( ( ord_less_nat @ M @ N )
              & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X ) )
            & ( ( F2 @ X )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9124_Taylor__down,axiom,
    ! [N: nat,Diff: nat > real > real,F2: real > real,A: real,B: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M: nat,T6: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C2 )
                  & ( ( F2 @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C2 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9125_Taylor__up,axiom,
    ! [N: nat,Diff: nat > real > real,F2: real > real,A: real,B: real,C2: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M: nat,T6: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C2 )
           => ( ( ord_less_real @ C2 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C2 @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F2 @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C2 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9126_Taylor,axiom,
    ! [N: nat,Diff: nat > real > real,F2: real > real,A: real,B: real,C2: real,X: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ( Diff @ zero_zero_nat )
          = F2 )
       => ( ! [M: nat,T6: real] :
              ( ( ( ord_less_nat @ M @ N )
                & ( ord_less_eq_real @ A @ T6 )
                & ( ord_less_eq_real @ T6 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C2 )
           => ( ( ord_less_eq_real @ C2 @ B )
             => ( ( ord_less_eq_real @ A @ X )
               => ( ( ord_less_eq_real @ X @ B )
                 => ( ( X != C2 )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ X @ T6 )
                            & ( ord_less_real @ T6 @ C2 ) ) )
                        & ( ~ ( ord_less_real @ X @ C2 )
                         => ( ( ord_less_real @ C2 @ T6 )
                            & ( ord_less_real @ T6 @ X ) ) )
                        & ( ( F2 @ X )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C2 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ M6 ) )
                              @ ( set_ord_lessThan_nat @ N ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N @ T6 ) @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ ( minus_minus_real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9127_Maclaurin__lemma2,axiom,
    ! [N: nat,H: real,Diff: nat > real > real,K2: nat,B5: real] :
      ( ! [M: nat,T6: real] :
          ( ( ( ord_less_nat @ M @ N )
            & ( ord_less_eq_real @ zero_zero_real @ T6 )
            & ( ord_less_eq_real @ T6 @ H ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
     => ( ( N
          = ( suc @ K2 ) )
       => ! [M3: nat,T7: real] :
            ( ( ( ord_less_nat @ M3 @ N )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ 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 @ B5 @ ( 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 ) @ T7 )
                @ ( 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 @ T7 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) )
                  @ ( times_times_real @ B5 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9128_DERIV__pow,axiom,
    ! [N: nat,X: real,S2: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X4: real] : ( power_power_real @ X4 @ N )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( power_power_real @ X @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X @ S2 ) ) ).

% DERIV_pow
thf(fact_9129_DERIV__const__ratio__const,axiom,
    ! [A: real,B: real,F2: real > real,K2: real] :
      ( ( A != B )
     => ( ! [X3: real] : ( has_fi5821293074295781190e_real @ F2 @ K2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
       => ( ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A ) )
          = ( times_times_real @ ( minus_minus_real @ B @ A ) @ K2 ) ) ) ) ).

% DERIV_const_ratio_const
thf(fact_9130_MVT2,axiom,
    ! [A: real,B: real,F2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
       => ? [Z: real] :
            ( ( ord_less_real @ A @ Z )
            & ( ord_less_real @ Z @ B )
            & ( ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z ) ) ) ) ) ) ).

% MVT2
thf(fact_9131_GMVT_H,axiom,
    ! [A: real,B: real,F2: real > real,G2: real > real,G3: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z: real] :
            ( ( ord_less_eq_real @ A @ Z )
           => ( ( ord_less_eq_real @ Z @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ F2 ) ) )
       => ( ! [Z: real] :
              ( ( ord_less_eq_real @ A @ Z )
             => ( ( ord_less_eq_real @ Z @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) @ G2 ) ) )
         => ( ! [Z: real] :
                ( ( ord_less_real @ A @ Z )
               => ( ( ord_less_real @ Z @ B )
                 => ( has_fi5821293074295781190e_real @ G2 @ ( G3 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
           => ( ! [Z: real] :
                  ( ( ord_less_real @ A @ Z )
                 => ( ( ord_less_real @ Z @ B )
                   => ( has_fi5821293074295781190e_real @ F2 @ ( F5 @ Z ) @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) )
             => ? [C: real] :
                  ( ( ord_less_real @ A @ C )
                  & ( ord_less_real @ C @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A ) ) @ ( G3 @ C ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G2 @ B ) @ ( G2 @ A ) ) @ ( F5 @ C ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9132_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 )
         => ! [N8: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9133_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 ) )
         => ! [N8: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9134_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9135_mult__nat__left__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat @ ( times_times_nat @ C2 ) @ at_top_nat @ at_top_nat ) ) ).

% mult_nat_left_at_top
thf(fact_9136_mult__nat__right__at__top,axiom,
    ! [C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C2 )
     => ( filterlim_nat_nat
        @ ^ [X4: nat] : ( times_times_nat @ X4 @ C2 )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9137_nested__sequence__unique,axiom,
    ! [F2: nat > real,G2: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( G2 @ ( suc @ N3 ) ) @ ( G2 @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F2 @ N3 ) @ ( G2 @ N3 ) )
         => ( ( filterlim_nat_real
              @ ^ [N4: nat] : ( minus_minus_real @ ( F2 @ N4 ) @ ( G2 @ N4 ) )
              @ ( topolo2815343760600316023s_real @ zero_zero_real )
              @ at_top_nat )
           => ? [L4: real] :
                ( ! [N8: nat] : ( ord_less_eq_real @ ( F2 @ N8 ) @ L4 )
                & ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N8: nat] : ( ord_less_eq_real @ L4 @ ( G2 @ N8 ) )
                & ( filterlim_nat_real @ G2 @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9138_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R4: real] :
        ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_real @ R4 @ ( X8 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( inverse_inverse_real @ ( X8 @ N4 ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9139_LIMSEQ__inverse__real__of__nat,axiom,
    ( filterlim_nat_real
    @ ^ [N4: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat
thf(fact_9140_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R3 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_9141_increasing__LIMSEQ,axiom,
    ! [F2: nat > real,L: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F2 @ N3 ) @ ( F2 @ ( suc @ N3 ) ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( F2 @ N3 ) @ L )
       => ( ! [E2: real] :
              ( ( ord_less_real @ zero_zero_real @ E2 )
             => ? [N8: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F2 @ N8 ) @ E2 ) ) )
         => ( filterlim_nat_real @ F2 @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9142_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( plus_plus_real @ R3 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_9143_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N4: nat] : ( times_times_real @ R3 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N4 ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_9144_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
          @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_9145_summable,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( summable_real
            @ ^ [N4: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N4 ) @ ( A @ N4 ) ) ) ) ) ) ).

% summable
thf(fact_9146_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J: nat] : ( minus_minus_real @ ( Theta @ J ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_9147_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J: nat] : ( cos_real @ ( Theta @ J ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K: nat > int] :
          ( filterlim_nat_real
          @ ^ [J: nat] : ( minus_minus_real @ ( Theta @ J ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K @ J ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_9148_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
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_9149_zeroseq__arctan__series,axiom,
    ! [X: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X ) @ one_one_real )
     => ( filterlim_nat_real
        @ ^ [N4: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ ( times_times_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% zeroseq_arctan_series
thf(fact_9150_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9151_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9152_sums__alternating__upper__lower,axiom,
    ! [A: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
         => ? [L4: real] :
              ( ! [N8: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N8: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N8 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N4: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9153_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
          @ ^ [N4: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9154_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N4: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I ) @ ( A @ I ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9155_real__bounded__linear,axiom,
    ( real_V5970128139526366754l_real
    = ( ^ [F4: real > real] :
        ? [C3: real] :
          ( F4
          = ( ^ [X4: real] : ( times_times_real @ X4 @ C3 ) ) ) ) ) ).

% real_bounded_linear
thf(fact_9156_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9157_tendsto__exp__limit__at__right,axiom,
    ! [X: real] :
      ( filterlim_real_real
      @ ^ [Y4: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X @ Y4 ) ) @ ( divide_divide_real @ one_one_real @ Y4 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_9158_GMVT,axiom,
    ! [A: real,B: real,F2: real > real,G2: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X3: real] :
            ( ( ( ord_less_eq_real @ A @ X3 )
              & ( ord_less_eq_real @ X3 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F2 ) )
       => ( ! [X3: real] :
              ( ( ( ord_less_real @ A @ X3 )
                & ( ord_less_real @ X3 @ B ) )
             => ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
         => ( ! [X3: real] :
                ( ( ( ord_less_eq_real @ A @ X3 )
                  & ( ord_less_eq_real @ X3 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G2 ) )
           => ( ! [X3: real] :
                  ( ( ( ord_less_real @ A @ X3 )
                    & ( ord_less_real @ X3 @ B ) )
                 => ( differ6690327859849518006l_real @ G2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C: real] :
                  ( ( has_fi5821293074295781190e_real @ G2 @ G_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F2 @ F_c @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C )
                  & ( ord_less_real @ C @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G2 @ B ) @ ( G2 @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_9159_eventually__sequentially__Suc,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat
        @ ^ [I: nat] : ( P2 @ ( suc @ I ) )
        @ at_top_nat )
      = ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9160_eventually__sequentially,axiom,
    ! [P2: nat > $o] :
      ( ( eventually_nat @ P2 @ at_top_nat )
      = ( ? [N5: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ N5 @ N4 )
           => ( P2 @ N4 ) ) ) ) ).

% eventually_sequentially
thf(fact_9161_eventually__sequentiallyI,axiom,
    ! [C2: nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( ord_less_eq_nat @ C2 @ X3 )
         => ( P2 @ X3 ) )
     => ( eventually_nat @ P2 @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9162_le__sequentially,axiom,
    ! [F3: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
      = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).

% le_sequentially
thf(fact_9163_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9164_atLeast__Suc__greaterThan,axiom,
    ! [K2: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K2 ) )
      = ( set_or1210151606488870762an_nat @ K2 ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9165_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9166_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_9167_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_9168_filterlim__pow__at__bot__even,axiom,
    ! [N: nat,F2: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F2 @ at_bot_real @ F3 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F2 @ X4 ) @ N )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9169_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9170_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_9171_GreatestI__nat,axiom,
    ! [P2: nat > $o,K2: nat,B: nat] :
      ( ( P2 @ K2 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_9172_filterlim__pow__at__bot__odd,axiom,
    ! [N: nat,F2: real > real,F3: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( filterlim_real_real @ F2 @ at_bot_real @ F3 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
         => ( filterlim_real_real
            @ ^ [X4: real] : ( power_power_real @ ( F2 @ X4 ) @ N )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9173_MVT,axiom,
    ! [A: real,B: real,F2: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F2 )
       => ( ! [X3: real] :
              ( ( ord_less_real @ A @ X3 )
             => ( ( ord_less_real @ X3 @ B )
               => ( differ6690327859849518006l_real @ F2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z: real] :
              ( ( ord_less_real @ A @ Z )
              & ( ord_less_real @ Z @ B )
              & ( has_fi5821293074295781190e_real @ F2 @ L4 @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F2 @ B ) @ ( F2 @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_9174_inj__sgn__power,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( inj_on_real_real
        @ ^ [Y4: real] : ( times_times_real @ ( sgn_sgn_real @ Y4 ) @ ( power_power_real @ ( abs_abs_real @ Y4 ) @ N ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9175_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9176_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9177_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_9178_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9179_inj__list__encode,axiom,
    ! [A4: set_list_nat] : ( inj_on_list_nat_nat @ nat_list_encode @ A4 ) ).

% inj_list_encode
thf(fact_9180_inj__list__decode,axiom,
    ! [A4: set_nat] : ( inj_on_nat_list_nat @ nat_list_decode @ A4 ) ).

% inj_list_decode
thf(fact_9181_inj__prod__encode,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] : ( inj_on2178005380612969504at_nat @ nat_prod_encode @ A4 ) ).

% inj_prod_encode
thf(fact_9182_inj__Suc,axiom,
    ! [N6: set_nat] : ( inj_on_nat_nat @ suc @ N6 ) ).

% inj_Suc
thf(fact_9183_inj__prod__decode,axiom,
    ! [A4: set_nat] : ( inj_on5538052773655684606at_nat @ nat_prod_decode @ A4 ) ).

% inj_prod_decode
thf(fact_9184_inj__on__diff__nat,axiom,
    ! [N6: set_nat,K2: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N6 )
         => ( ord_less_eq_nat @ K2 @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N4: nat] : ( minus_minus_nat @ N4 @ K2 )
        @ N6 ) ) ).

% inj_on_diff_nat
thf(fact_9185_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_9186_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_9187_Gcd__eq__Max,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ( gcd_Gcd_nat @ M5 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M6: nat] :
                      ( collect_nat
                      @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ M6 ) )
                  @ M5 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9188_Max__divisors__self__nat,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] : ( dvd_dvd_nat @ D2 @ N ) ) )
        = N ) ) ).

% Max_divisors_self_nat
thf(fact_9189_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X7: set_nat] : ( if_nat @ ( X7 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X7 ) ) ) ) ).

% Sup_nat_def
thf(fact_9190_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_9191_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N4: nat] :
          ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N4 ) @ M6 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9192_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9193_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ~ ( ( Deg2 = Xa2 )
                  & ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                     => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                  & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( case_o184042715313410164at_nat
                    @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I: nat] :
                              ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X4: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
                                   => ( ( ord_less_nat @ Mi3 @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9194_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( Deg2 = Xa2 )
                & ! [X3: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                & ( case_o184042715313410164at_nat
                  @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                    & ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I: nat] :
                            ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
                                 => ( ( ord_less_nat @ Mi3 @ X4 )
                                    & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9195_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg3: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg3 )
      = ( ( Deg = Deg3 )
        & ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( 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 @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
            & ! [X4: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I ) @ X7 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ 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 ) ) ) @ TreeList2 @ X4 )
                         => ( ( ord_less_nat @ Mi3 @ X4 )
                            & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9196_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ( ( Deg2 = Xa2 )
                    & ! [X3: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ TreeList ) )
                       => ( vEBT_VEBT_valid @ X3 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( case_o184042715313410164at_nat
                      @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                        & ! [X4: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                           => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I: nat] :
                                ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X4: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                 => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
                                     => ( ( ord_less_nat @ Mi3 @ X4 )
                                        & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9197_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
               => ( Xa2 != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) @ Xa2 ) )
                 => ~ ( ( Deg2 = Xa2 )
                      & ! [X5: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ X4 )
                                       => ( ( ord_less_nat @ Mi3 @ X4 )
                                          & ( ord_less_eq_nat @ X4 @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9198_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X @ Xa2 )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa2 = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa2 )
                      & ! [X4: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
                          & ! [X4: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                             => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I: nat] :
                                  ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I ) @ X7 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X4: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList ) )
                                   => ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X7 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList @ 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 ) ) ) @ TreeList @ 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 @ TreeList @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9199_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_9200_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9201_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9202_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_9203_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_9204_take__bit__num__simps_I3_J,axiom,
    ! [N: nat,M2: num] :
      ( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N @ M2 ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9205_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_9206_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_9207_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_9208_min__diff,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I2 ) ) ).

% min_diff
thf(fact_9209_nat__mult__min__left,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M2 @ N ) @ Q2 )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ Q2 ) @ ( times_times_nat @ N @ Q2 ) ) ) ).

% nat_mult_min_left
thf(fact_9210_nat__mult__min__right,axiom,
    ! [M2: nat,N: nat,Q2: nat] :
      ( ( times_times_nat @ M2 @ ( ord_min_nat @ N @ Q2 ) )
      = ( ord_min_nat @ ( times_times_nat @ M2 @ N ) @ ( times_times_nat @ M2 @ Q2 ) ) ) ).

% nat_mult_min_right
thf(fact_9211_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9212_min__Suc1,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_min_nat @ ( suc @ N ) @ M2 )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ N @ M7 ) )
        @ M2 ) ) ).

% min_Suc1
thf(fact_9213_min__Suc2,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ M2 @ ( suc @ N ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M7: nat] : ( suc @ ( ord_min_nat @ M7 @ N ) )
        @ M2 ) ) ).

% min_Suc2
thf(fact_9214_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N4: nat,M6: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M6 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N4 @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9215_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M6: nat,N4: nat] :
            ( N4
            = ( suc @ M6 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9216_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I: int,N4: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I ) @ ( semiri5074537144036343181t_real @ N4 ) ) )
          & ( N4 != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9217_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_9218_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
                  @ ^ [R: code_integer,S6: code_integer] : ( if_Pro6119634080678213985nteger @ ( S6 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S6 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9219_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_9220_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_9221_bezw__aux,axiom,
    ! [X: nat,Y: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X @ Y ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X @ Y ) ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ) ).

% bezw_aux
thf(fact_9222_gcd__0__left__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X )
      = X ) ).

% gcd_0_left_nat
thf(fact_9223_gcd__0__nat,axiom,
    ! [X: nat] :
      ( ( gcd_gcd_nat @ X @ zero_zero_nat )
      = X ) ).

% gcd_0_nat
thf(fact_9224_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9225_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9226_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9227_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9228_gcd__Suc__0,axiom,
    ! [M2: nat] :
      ( ( gcd_gcd_nat @ M2 @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9229_gcd__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( ( M2 != zero_zero_nat )
        | ( N != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9230_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_9231_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_9232_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_9233_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_9234_gcd__mult__distrib__nat,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( times_times_nat @ K2 @ ( gcd_gcd_nat @ M2 @ N ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9235_gcd__non__0__nat,axiom,
    ! [Y: nat,X: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9236_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X4: nat,Y4: nat] : ( if_nat @ ( Y4 = zero_zero_nat ) @ X4 @ ( gcd_gcd_nat @ Y4 @ ( modulo_modulo_nat @ X4 @ Y4 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9237_gcd__nat_Oelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y = X ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9238_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X3: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X3 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9239_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X3: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X3 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X3 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X3 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9240_Field__natLeq__on,axiom,
    ! [N: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y4 @ N )
                & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).

% Field_natLeq_on
thf(fact_9241_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M6: nat,N4: nat] :
        ( ( dvd_dvd_nat @ M6 @ N4 )
        & ( M6 != N4 ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9242_gcd__is__Max__divisors__nat,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( gcd_gcd_nat @ M2 @ N )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D2: nat] :
                ( ( dvd_dvd_nat @ D2 @ M2 )
                & ( dvd_dvd_nat @ D2 @ N ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9243_gcd__nat_Opelims,axiom,
    ! [X: nat,Xa2: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X @ Xa2 )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y = X ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9244_gcd__mult__distrib__int,axiom,
    ! [K2: int,M2: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ K2 ) @ ( gcd_gcd_int @ M2 @ N ) )
      = ( gcd_gcd_int @ ( times_times_int @ K2 @ M2 ) @ ( times_times_int @ K2 @ N ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9245_bezout__int,axiom,
    ! [X: int,Y: int] :
    ? [U3: int,V: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X ) @ ( times_times_int @ V @ Y ) )
      = ( gcd_gcd_int @ X @ Y ) ) ).

% bezout_int
thf(fact_9246_strict__mono__imp__increasing,axiom,
    ! [F2: nat > nat,N: nat] :
      ( ( order_5726023648592871131at_nat @ F2 )
     => ( ord_less_eq_nat @ N @ ( F2 @ N ) ) ) ).

% strict_mono_imp_increasing
thf(fact_9247_DeMoivre2,axiom,
    ! [R3: real,A: real,N: nat] :
      ( ( power_power_complex @ ( rcis @ R3 @ A ) @ N )
      = ( rcis @ ( power_power_real @ R3 @ N ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ A ) ) ) ).

% DeMoivre2
thf(fact_9248_Re__rcis,axiom,
    ! [R3: real,A: real] :
      ( ( re @ ( rcis @ R3 @ A ) )
      = ( times_times_real @ R3 @ ( cos_real @ A ) ) ) ).

% Re_rcis
thf(fact_9249_Im__rcis,axiom,
    ! [R3: real,A: real] :
      ( ( im @ ( rcis @ R3 @ A ) )
      = ( times_times_real @ R3 @ ( sin_real @ A ) ) ) ).

% Im_rcis
thf(fact_9250_rcis__mult,axiom,
    ! [R1: real,A: real,R22: real,B: real] :
      ( ( times_times_complex @ ( rcis @ R1 @ A ) @ ( rcis @ R22 @ B ) )
      = ( rcis @ ( times_times_real @ R1 @ R22 ) @ ( plus_plus_real @ A @ B ) ) ) ).

% rcis_mult
thf(fact_9251_rcis__def,axiom,
    ( rcis
    = ( ^ [R: real,A5: real] : ( times_times_complex @ ( real_V4546457046886955230omplex @ R ) @ ( cis @ A5 ) ) ) ) ).

% rcis_def
thf(fact_9252_eventually__prod__sequentially,axiom,
    ! [P2: product_prod_nat_nat > $o] :
      ( ( eventu1038000079068216329at_nat @ P2 @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
      = ( ? [N5: nat] :
          ! [M6: nat] :
            ( ( ord_less_eq_nat @ N5 @ M6 )
           => ! [N4: nat] :
                ( ( ord_less_eq_nat @ N5 @ N4 )
               => ( P2 @ ( product_Pair_nat_nat @ N4 @ M6 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_9253_sqr_Osimps_I3_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit1 @ N ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N ) @ N ) ) ) ) ).

% sqr.simps(3)
thf(fact_9254_sqr_Osimps_I2_J,axiom,
    ! [N: num] :
      ( ( sqr @ ( bit0 @ N ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).

% sqr.simps(2)
thf(fact_9255_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9256_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X4: num] : ( times_times_num @ X4 @ X4 ) ) ) ).

% sqr_conv_mult
thf(fact_9257_pow_Osimps_I3_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X @ Y ) ) @ X ) ) ).

% pow.simps(3)
thf(fact_9258_pow_Osimps_I1_J,axiom,
    ! [X: num] :
      ( ( pow @ X @ one )
      = X ) ).

% pow.simps(1)
thf(fact_9259_pow_Osimps_I2_J,axiom,
    ! [X: num,Y: num] :
      ( ( pow @ X @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9260_natLeq__on__wo__rel,axiom,
    ! [N: nat] :
      ( bNF_We3818239936649020644el_nat
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y4 @ N )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_wo_rel
thf(fact_9261_pairs__le__eq__Sigma,axiom,
    ! [M2: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I: nat,J: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I @ J ) @ M2 ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M2 )
        @ ^ [R: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M2 @ R ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_9262_finite__vimage__Suc__iff,axiom,
    ! [F3: set_nat] :
      ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
      = ( finite_finite_nat @ F3 ) ) ).

% finite_vimage_Suc_iff
thf(fact_9263_coprime__Suc__0__right,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ zero_zero_nat ) ) ).

% coprime_Suc_0_right
thf(fact_9264_coprime__Suc__0__left,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ zero_zero_nat ) @ N ) ).

% coprime_Suc_0_left
thf(fact_9265_coprime__Suc__left__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ ( suc @ N ) @ N ) ).

% coprime_Suc_left_nat
thf(fact_9266_coprime__Suc__right__nat,axiom,
    ! [N: nat] : ( algebr934650988132801477me_nat @ N @ ( suc @ N ) ) ).

% coprime_Suc_right_nat
thf(fact_9267_coprime__crossproduct__nat,axiom,
    ! [A: nat,D: nat,B: nat,C2: nat] :
      ( ( algebr934650988132801477me_nat @ A @ D )
     => ( ( algebr934650988132801477me_nat @ B @ C2 )
       => ( ( ( times_times_nat @ A @ C2 )
            = ( times_times_nat @ B @ D ) )
          = ( ( A = B )
            & ( C2 = D ) ) ) ) ) ).

% coprime_crossproduct_nat
thf(fact_9268_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_9269_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_9270_coprime__diff__one__left__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ N ) ) ).

% coprime_diff_one_left_nat
thf(fact_9271_coprime__diff__one__right__nat,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( algebr934650988132801477me_nat @ N @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ).

% coprime_diff_one_right_nat
thf(fact_9272_Rats__abs__nat__div__natE,axiom,
    ! [X: real] :
      ( ( member_real @ X @ field_5140801741446780682s_real )
     => ~ ! [M: nat,N3: nat] :
            ( ( N3 != zero_zero_nat )
           => ( ( ( abs_abs_real @ X )
                = ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N3 ) ) )
             => ~ ( algebr934650988132801477me_nat @ M @ N3 ) ) ) ) ).

% Rats_abs_nat_div_natE
thf(fact_9273_set__decode__div__2,axiom,
    ! [X: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X ) ) ) ).

% set_decode_div_2
thf(fact_9274_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_9275_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,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y4 @ N )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% Restr_natLeq
thf(fact_9276_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9277_coprime__crossproduct__int,axiom,
    ! [A: int,D: int,B: int,C2: int] :
      ( ( algebr932160517623751201me_int @ A @ D )
     => ( ( algebr932160517623751201me_int @ B @ C2 )
       => ( ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ C2 ) )
            = ( times_times_int @ ( abs_abs_int @ B ) @ ( abs_abs_int @ D ) ) )
          = ( ( ( abs_abs_int @ A )
              = ( abs_abs_int @ B ) )
            & ( ( abs_abs_int @ C2 )
              = ( abs_abs_int @ D ) ) ) ) ) ) ).

% coprime_crossproduct_int
thf(fact_9278_natLeq__Linear__order,axiom,
    order_4473980167227706203on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Linear_order
thf(fact_9279_natLeq__Total,axiom,
    total_on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Total
thf(fact_9280_natLeq__Refl,axiom,
    refl_on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Refl
thf(fact_9281_natLeq__natLess__Id,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( minus_1356011639430497352at_nat @ bNF_Ca8665028551170535155natLeq @ id_nat2 ) ) ).

% natLeq_natLess_Id
thf(fact_9282_Field__natLeq,axiom,
    ( ( field_nat @ bNF_Ca8665028551170535155natLeq )
    = top_top_set_nat ) ).

% Field_natLeq
thf(fact_9283_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_9284_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,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y4 @ N )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_9285_natLeq__underS__less,axiom,
    ! [N: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N )
      = ( collect_nat
        @ ^ [X4: nat] : ( ord_less_nat @ X4 @ N ) ) ) ).

% natLeq_underS_less
thf(fact_9286_pair__lessI2,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_9287_pair__less__iff1,axiom,
    ! [X: nat,Y: nat,Z3: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ X @ Z3 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y @ Z3 ) ) ).

% pair_less_iff1
thf(fact_9288_pair__lessI1,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_9289_pair__leqI2,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S2 @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_9290_pair__leqI1,axiom,
    ! [A: nat,B: nat,S2: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S2 ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_9291_length__upto,axiom,
    ! [I2: int,J2: int] :
      ( ( size_size_list_int @ ( upto @ I2 @ J2 ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J2 @ I2 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9292_gcd__nat_Oordering__top__axioms,axiom,
    ( ordering_top_nat @ dvd_dvd_nat
    @ ^ [M6: nat,N4: nat] :
        ( ( dvd_dvd_nat @ M6 @ N4 )
        & ( M6 != N4 ) )
    @ zero_zero_nat ) ).

% gcd_nat.ordering_top_axioms
thf(fact_9293_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X4: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X4 )
    @ ^ [X4: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X4 )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_9294_ratrel__iff,axiom,
    ( ratrel
    = ( ^ [X4: product_prod_int_int,Y4: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X4 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y4 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).

% ratrel_iff
thf(fact_9295_ratrel__def,axiom,
    ( ratrel
    = ( ^ [X4: product_prod_int_int,Y4: product_prod_int_int] :
          ( ( ( product_snd_int_int @ X4 )
           != zero_zero_int )
          & ( ( product_snd_int_int @ Y4 )
           != zero_zero_int )
          & ( ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) )
            = ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ) ).

% ratrel_def
thf(fact_9296_plus__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X @ X )
       => ( ( plus_plus_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% plus_rat.abs_eq
thf(fact_9297_times__rat_Oabs__eq,axiom,
    ! [Xa2: product_prod_int_int,X: product_prod_int_int] :
      ( ( ratrel @ Xa2 @ Xa2 )
     => ( ( ratrel @ X @ X )
       => ( ( times_times_rat @ ( abs_Rat @ Xa2 ) @ ( abs_Rat @ X ) )
          = ( abs_Rat @ ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ Xa2 ) @ ( product_fst_int_int @ X ) ) @ ( times_times_int @ ( product_snd_int_int @ Xa2 ) @ ( product_snd_int_int @ X ) ) ) ) ) ) ) ).

% times_rat.abs_eq
thf(fact_9298_Rat_Opositive_Oabs__eq,axiom,
    ! [X: product_prod_int_int] :
      ( ( ratrel @ X @ X )
     => ( ( positive @ ( abs_Rat @ X ) )
        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive.abs_eq
thf(fact_9299_Rat_Opositive__mult,axiom,
    ! [X: rat,Y: rat] :
      ( ( positive @ X )
     => ( ( positive @ Y )
       => ( positive @ ( times_times_rat @ X @ Y ) ) ) ) ).

% Rat.positive_mult
thf(fact_9300_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X4: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X4 ) ) @ ( product_snd_int_int @ ( rep_Rat @ X4 ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9301_Rat_Opositive_Orsp,axiom,
    ( bNF_re8699439704749558557nt_o_o @ ratrel
    @ ^ [Y5: $o,Z2: $o] : Y5 = Z2
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ).

% Rat.positive.rsp
thf(fact_9302_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) ) ) ) ).

% Rat.positive_def
thf(fact_9303_less__eq__natural_Orsp,axiom,
    ( bNF_re578469030762574527_nat_o
    @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
    @ ( bNF_re4705727531993890431at_o_o
      @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
      @ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
    @ ord_less_eq_nat
    @ ord_less_eq_nat ) ).

% less_eq_natural.rsp
thf(fact_9304_Suc_Orsp,axiom,
    ( bNF_re5653821019739307937at_nat
    @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
    @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
    @ suc
    @ suc ) ).

% Suc.rsp
thf(fact_9305_times__integer_Orsp,axiom,
    ( bNF_re711492959462206631nt_int
    @ ^ [Y5: int,Z2: int] : Y5 = Z2
    @ ( bNF_re4712519889275205905nt_int
      @ ^ [Y5: int,Z2: int] : Y5 = Z2
      @ ^ [Y5: int,Z2: int] : Y5 = Z2 )
    @ times_times_int
    @ times_times_int ) ).

% times_integer.rsp
thf(fact_9306_times__natural_Orsp,axiom,
    ( bNF_re1345281282404953727at_nat
    @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
    @ ( bNF_re5653821019739307937at_nat
      @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
      @ ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
    @ times_times_nat
    @ times_times_nat ) ).

% times_natural.rsp
thf(fact_9307_times__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ).

% times_rat.rsp
thf(fact_9308_plus__rat_Orsp,axiom,
    ( bNF_re5228765855967844073nt_int @ ratrel @ ( bNF_re7145576690424134365nt_int @ ratrel @ ratrel )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ).

% plus_rat.rsp
thf(fact_9309_plus__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ plus_plus_rat ) ).

% plus_rat.transfer
thf(fact_9310_Rat_Opositive_Otransfer,axiom,
    ( bNF_re1494630372529172596at_o_o @ pcr_rat
    @ ^ [Y5: $o,Z2: $o] : Y5 = Z2
    @ ^ [X4: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ X4 ) ) )
    @ positive ) ).

% Rat.positive.transfer
thf(fact_9311_times__rat_Otransfer,axiom,
    ( bNF_re7627151682743391978at_rat @ pcr_rat @ ( bNF_re8279943556446156061nt_rat @ pcr_rat @ pcr_rat )
    @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) )
    @ times_times_rat ) ).

% times_rat.transfer
thf(fact_9312_times__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
    @ times_times_int ) ).

% times_int.transfer
thf(fact_9313_minus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
    @ minus_minus_int ) ).

% minus_int.transfer
thf(fact_9314_zero__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) @ zero_zero_int ).

% zero_int.transfer
thf(fact_9315_int__transfer,axiom,
    ( bNF_re6830278522597306478at_int
    @ ^ [Y5: nat,Z2: nat] : Y5 = Z2
    @ pcr_int
    @ ^ [N4: nat] : ( product_Pair_nat_nat @ N4 @ zero_zero_nat )
    @ semiri1314217659103216013at_int ) ).

% int_transfer
thf(fact_9316_uminus__int_Otransfer,axiom,
    ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) )
    @ uminus_uminus_int ) ).

% uminus_int.transfer
thf(fact_9317_one__int_Otransfer,axiom,
    pcr_int @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) @ one_one_int ).

% one_int.transfer
thf(fact_9318_less__eq__int_Otransfer,axiom,
    ( bNF_re717283939379294677_int_o @ pcr_int
    @ ( bNF_re6644619430987730960nt_o_o @ pcr_int
      @ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ord_less_eq_int ) ).

% less_eq_int.transfer
thf(fact_9319_plus__int_Otransfer,axiom,
    ( bNF_re7408651293131936558nt_int @ pcr_int @ ( bNF_re7400052026677387805at_int @ pcr_int @ pcr_int )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V3 ) ) ) )
    @ plus_plus_int ) ).

% plus_int.transfer
thf(fact_9320_times__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X4 @ U2 ) @ ( times_times_nat @ Y4 @ V3 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X4 @ V3 ) @ ( times_times_nat @ Y4 @ U2 ) ) ) ) ) ) ).

% times_int.rsp
thf(fact_9321_minus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ Y4 @ U2 ) ) ) ) ) ).

% minus_int.rsp
thf(fact_9322_intrel__iff,axiom,
    ! [X: nat,Y: nat,U: nat,V2: nat] :
      ( ( intrel @ ( product_Pair_nat_nat @ X @ Y ) @ ( product_Pair_nat_nat @ U @ V2 ) )
      = ( ( plus_plus_nat @ X @ V2 )
        = ( plus_plus_nat @ U @ Y ) ) ) ).

% intrel_iff
thf(fact_9323_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_9324_uminus__int_Orsp,axiom,
    ( bNF_re2241393799969408733at_nat @ intrel @ intrel
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) )
    @ ( produc2626176000494625587at_nat
      @ ^ [X4: nat,Y4: nat] : ( product_Pair_nat_nat @ Y4 @ X4 ) ) ) ).

% uminus_int.rsp
thf(fact_9325_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_9326_less__eq__int_Orsp,axiom,
    ( bNF_re4202695980764964119_nat_o @ intrel
    @ ( bNF_re3666534408544137501at_o_o @ intrel
      @ ^ [Y5: $o,Z2: $o] : Y5 = Z2 )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) )
    @ ( produc8739625826339149834_nat_o
      @ ^ [X4: nat,Y4: nat] :
          ( produc6081775807080527818_nat_o
          @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ).

% less_eq_int.rsp
thf(fact_9327_plus__int_Orsp,axiom,
    ( bNF_re3099431351363272937at_nat @ intrel @ ( bNF_re2241393799969408733at_nat @ intrel @ intrel )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V3 ) ) ) )
    @ ( produc27273713700761075at_nat
      @ ^ [X4: nat,Y4: nat] :
          ( produc2626176000494625587at_nat
          @ ^ [U2: nat,V3: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X4 @ U2 ) @ ( plus_plus_nat @ Y4 @ V3 ) ) ) ) ) ).

% plus_int.rsp
thf(fact_9328_less__eq__enat__def,axiom,
    ( ord_le2932123472753598470d_enat
    = ( ^ [M6: extended_enat] :
          ( extended_case_enat_o
          @ ^ [N1: nat] :
              ( extended_case_enat_o
              @ ^ [M1: nat] : ( ord_less_eq_nat @ M1 @ N1 )
              @ $false
              @ M6 )
          @ $true ) ) ) ).

% less_eq_enat_def
thf(fact_9329_natLeq__Well__order,axiom,
    order_2888998067076097458on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Well_order
thf(fact_9330_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,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y4 @ N )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_well_order_on
thf(fact_9331_natLeq__on__Well__order,axiom,
    ! [N: nat] :
      ( order_2888998067076097458on_nat
      @ ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X4: nat,Y4: nat] :
                ( ( ord_less_nat @ X4 @ N )
                & ( ord_less_nat @ Y4 @ N )
                & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) )
      @ ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X4: nat,Y4: nat] :
              ( ( ord_less_nat @ X4 @ N )
              & ( ord_less_nat @ Y4 @ N )
              & ( ord_less_eq_nat @ X4 @ Y4 ) ) ) ) ) ).

% natLeq_on_Well_order
thf(fact_9332_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_9333_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_9334_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_9335_plus__rat__def,axiom,
    ( plus_plus_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) @ ( times_times_int @ ( product_fst_int_int @ Y4 ) @ ( product_snd_int_int @ X4 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ) ).

% plus_rat_def
thf(fact_9336_diff__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( minus_minus_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
          = ( fract @ ( minus_minus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C2 @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% diff_rat
thf(fact_9337_sgn__rat,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_rat @ ( fract @ A @ B ) )
      = ( ring_1_of_int_rat @ ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ) ).

% sgn_rat
thf(fact_9338_mult__rat,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( times_times_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
      = ( fract @ ( times_times_int @ A @ C2 ) @ ( times_times_int @ B @ D ) ) ) ).

% mult_rat
thf(fact_9339_divide__rat,axiom,
    ! [A: int,B: int,C2: int,D: int] :
      ( ( divide_divide_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
      = ( fract @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C2 ) ) ) ).

% divide_rat
thf(fact_9340_less__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9341_add__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( plus_plus_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
          = ( fract @ ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ C2 @ B ) ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% add_rat
thf(fact_9342_le__rat,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C2 @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C2 @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_9343_mult__rat__cancel,axiom,
    ! [C2: int,A: int,B: int] :
      ( ( C2 != zero_zero_int )
     => ( ( fract @ ( times_times_int @ C2 @ A ) @ ( times_times_int @ C2 @ B ) )
        = ( fract @ A @ B ) ) ) ).

% mult_rat_cancel
thf(fact_9344_eq__rat_I1_J,axiom,
    ! [B: int,D: int,A: int,C2: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ( fract @ A @ B )
            = ( fract @ C2 @ D ) )
          = ( ( times_times_int @ A @ D )
            = ( times_times_int @ C2 @ B ) ) ) ) ) ).

% eq_rat(1)
thf(fact_9345_positive__rat,axiom,
    ! [A: int,B: int] :
      ( ( positive @ ( fract @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% positive_rat
thf(fact_9346_times__rat__def,axiom,
    ( times_times_rat
    = ( map_fu4333342158222067775at_rat @ rep_Rat @ ( map_fu5673905371560938248nt_rat @ rep_Rat @ abs_Rat )
      @ ^ [X4: product_prod_int_int,Y4: product_prod_int_int] : ( product_Pair_int_int @ ( times_times_int @ ( product_fst_int_int @ X4 ) @ ( product_fst_int_int @ Y4 ) ) @ ( times_times_int @ ( product_snd_int_int @ X4 ) @ ( product_snd_int_int @ Y4 ) ) ) ) ) ).

% times_rat_def
thf(fact_9347_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_9348_char_Osize_I2_J,axiom,
    ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size(2)
thf(fact_9349_char_Osize__gen,axiom,
    ! [X1: $o,X2: $o,X33: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
      ( ( size_char @ ( char2 @ X1 @ X2 @ X33 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
      = zero_zero_nat ) ).

% char.size_gen
thf(fact_9350_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_9351_idiff__enat__0__right,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N @ ( extended_enat2 @ zero_zero_nat ) )
      = N ) ).

% idiff_enat_0_right
thf(fact_9352_idiff__enat__0,axiom,
    ! [N: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ ( extended_enat2 @ zero_zero_nat ) @ N )
      = ( extended_enat2 @ zero_zero_nat ) ) ).

% idiff_enat_0
thf(fact_9353_times__enat__simps_I1_J,axiom,
    ! [M2: nat,N: nat] :
      ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( times_times_nat @ M2 @ N ) ) ) ).

% times_enat_simps(1)
thf(fact_9354_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_9355_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_9356_enat__0__iff_I2_J,axiom,
    ! [X: nat] :
      ( ( zero_z5237406670263579293d_enat
        = ( extended_enat2 @ X ) )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(2)
thf(fact_9357_enat__0__iff_I1_J,axiom,
    ! [X: nat] :
      ( ( ( extended_enat2 @ X )
        = zero_z5237406670263579293d_enat )
      = ( X = zero_zero_nat ) ) ).

% enat_0_iff(1)
thf(fact_9358_zero__enat__def,axiom,
    ( zero_z5237406670263579293d_enat
    = ( extended_enat2 @ zero_zero_nat ) ) ).

% zero_enat_def
thf(fact_9359_iadd__le__enat__iff,axiom,
    ! [X: extended_enat,Y: extended_enat,N: nat] :
      ( ( ord_le2932123472753598470d_enat @ ( plus_p3455044024723400733d_enat @ X @ Y ) @ ( extended_enat2 @ N ) )
      = ( ? [Y8: nat,X9: nat] :
            ( ( X
              = ( extended_enat2 @ X9 ) )
            & ( Y
              = ( extended_enat2 @ Y8 ) )
            & ( ord_less_eq_nat @ ( plus_plus_nat @ X9 @ Y8 ) @ N ) ) ) ) ).

% iadd_le_enat_iff
thf(fact_9360_elimnum,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
        = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimnum
thf(fact_9361_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ ( extended_enat2 @ L ) )
      = ( vEBT_Node @ Info2 @ 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 ) ) ) ) ) )
            @ TreeList2 ) )
        @ ( 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_9362_enat__0__less__mult__iff,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M2 @ N ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M2 )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N ) ) ) ).

% enat_0_less_mult_iff
thf(fact_9363_imult__is__0,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M2 @ N )
        = zero_z5237406670263579293d_enat )
      = ( ( M2 = zero_z5237406670263579293d_enat )
        | ( N = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_9364_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,Uu: extended_enat] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A @ B ) @ Uu )
      = ( vEBT_Leaf @ A @ B ) ) ).

% VEBT_internal.elim_dead.simps(1)
thf(fact_9365_VEBT__internal_Oelim__dead_Oelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ! [A3: $o,B3: $o] :
            ( ( X
              = ( vEBT_Leaf @ A3 @ B3 ) )
           => ( Y
             != ( vEBT_Leaf @ A3 @ B3 ) ) )
       => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X
                = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
             => ( ( Xa2 = extend5688581933313929465d_enat )
               => ( Y
                 != ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
                      @ TreeList )
                    @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) ) ) )
         => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ! [L4: nat] :
                    ( ( Xa2
                      = ( extended_enat2 @ L4 ) )
                   => ( Y
                     != ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
                            @ TreeList ) )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( 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_9366_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
      ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
      = ( vEBT_Node @ Info2 @ 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 ) ) ) ) ) )
          @ TreeList2 )
        @ ( vEBT_VEBT_elim_dead @ Summary @ extend5688581933313929465d_enat ) ) ) ).

% VEBT_internal.elim_dead.simps(2)
thf(fact_9367_elimcomplete,axiom,
    ! [Info2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ N )
     => ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) @ extend5688581933313929465d_enat )
        = ( vEBT_Node @ Info2 @ Deg @ TreeList2 @ Summary ) ) ) ).

% elimcomplete
thf(fact_9368_times__enat__simps_I2_J,axiom,
    ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ extend5688581933313929465d_enat )
    = extend5688581933313929465d_enat ) ).

% times_enat_simps(2)
thf(fact_9369_times__enat__simps_I3_J,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = zero_z5237406670263579293d_enat ) )
      & ( ( N != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ ( extended_enat2 @ N ) )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(3)
thf(fact_9370_times__enat__simps_I4_J,axiom,
    ! [M2: nat] :
      ( ( ( M2 = zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M2 ) @ extend5688581933313929465d_enat )
          = zero_z5237406670263579293d_enat ) )
      & ( ( M2 != zero_zero_nat )
       => ( ( times_7803423173614009249d_enat @ ( extended_enat2 @ M2 ) @ extend5688581933313929465d_enat )
          = extend5688581933313929465d_enat ) ) ) ).

% times_enat_simps(4)
thf(fact_9371_imult__is__infinity,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ A @ B )
        = extend5688581933313929465d_enat )
      = ( ( ( A = extend5688581933313929465d_enat )
          & ( B != zero_z5237406670263579293d_enat ) )
        | ( ( B = extend5688581933313929465d_enat )
          & ( A != zero_z5237406670263579293d_enat ) ) ) ) ).

% imult_is_infinity
thf(fact_9372_VEBT__internal_Oelim__dead_Ocases,axiom,
    ! [X: produc7272778201969148633d_enat] :
      ( ! [A3: $o,B3: $o,Uu2: extended_enat] :
          ( X
         != ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A3 @ B3 ) @ Uu2 ) )
     => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
            ( X
           != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) )
       => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT,L4: nat] :
              ( X
             != ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ).

% VEBT_internal.elim_dead.cases
thf(fact_9373_imult__infinity,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
     => ( ( times_7803423173614009249d_enat @ extend5688581933313929465d_enat @ N )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity
thf(fact_9374_imult__infinity__right,axiom,
    ! [N: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N )
     => ( ( times_7803423173614009249d_enat @ N @ extend5688581933313929465d_enat )
        = extend5688581933313929465d_enat ) ) ).

% imult_infinity_right
thf(fact_9375_times__enat__def,axiom,
    ( times_7803423173614009249d_enat
    = ( ^ [M6: extended_enat,N4: extended_enat] :
          ( extend3600170679010898289d_enat
          @ ^ [O: nat] :
              ( extend3600170679010898289d_enat
              @ ^ [P5: nat] : ( extended_enat2 @ ( times_times_nat @ O @ P5 ) )
              @ ( if_Extended_enat @ ( O = zero_zero_nat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
              @ N4 )
          @ ( if_Extended_enat @ ( N4 = zero_z5237406670263579293d_enat ) @ zero_z5237406670263579293d_enat @ extend5688581933313929465d_enat )
          @ M6 ) ) ) ).

% times_enat_def
thf(fact_9376_VEBT__internal_Oelim__dead_Opelims,axiom,
    ! [X: vEBT_VEBT,Xa2: extended_enat,Y: vEBT_VEBT] :
      ( ( ( vEBT_VEBT_elim_dead @ X @ Xa2 )
        = Y )
     => ( ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ X @ Xa2 ) )
       => ( ! [A3: $o,B3: $o] :
              ( ( X
                = ( vEBT_Leaf @ A3 @ B3 ) )
             => ( ( Y
                  = ( vEBT_Leaf @ A3 @ B3 ) )
               => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Leaf @ A3 @ B3 ) @ Xa2 ) ) ) )
         => ( ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X
                  = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
               => ( ( Xa2 = extend5688581933313929465d_enat )
                 => ( ( Y
                      = ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
                          @ TreeList )
                        @ ( vEBT_VEBT_elim_dead @ Summary2 @ extend5688581933313929465d_enat ) ) )
                   => ~ ( accp_P6183159247885693666d_enat @ vEBT_V312737461966249ad_rel @ ( produc581526299967858633d_enat @ ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) @ extend5688581933313929465d_enat ) ) ) ) )
           => ~ ! [Info: option4927543243414619207at_nat,Deg2: nat,TreeList: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                  ( ( X
                    = ( vEBT_Node @ Info @ Deg2 @ TreeList @ Summary2 ) )
                 => ! [L4: nat] :
                      ( ( Xa2
                        = ( extended_enat2 @ L4 ) )
                     => ( ( Y
                          = ( vEBT_Node @ Info @ 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 ) ) ) ) ) )
                                @ TreeList ) )
                            @ ( vEBT_VEBT_elim_dead @ Summary2 @ ( 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 @ Info @ Deg2 @ TreeList @ Summary2 ) @ ( extended_enat2 @ L4 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.elim_dead.pelims
thf(fact_9377_eSuc__def,axiom,
    ( extended_eSuc
    = ( extend3600170679010898289d_enat
      @ ^ [N4: nat] : ( extended_enat2 @ ( suc @ N4 ) )
      @ extend5688581933313929465d_enat ) ) ).

% eSuc_def
thf(fact_9378_binomial__def,axiom,
    ( binomial
    = ( ^ [N4: nat,K3: nat] :
          ( finite_card_set_nat
          @ ( collect_set_nat
            @ ^ [K7: set_nat] :
                ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N4 ) ) )
                & ( ( finite_card_nat @ K7 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9379_enat__eSuc__iff,axiom,
    ! [Y: nat,X: extended_enat] :
      ( ( ( extended_enat2 @ Y )
        = ( extended_eSuc @ X ) )
      = ( ? [N4: nat] :
            ( ( Y
              = ( suc @ N4 ) )
            & ( ( extended_enat2 @ N4 )
              = X ) ) ) ) ).

% enat_eSuc_iff
thf(fact_9380_eSuc__enat__iff,axiom,
    ! [X: extended_enat,Y: nat] :
      ( ( ( extended_eSuc @ X )
        = ( extended_enat2 @ Y ) )
      = ( ? [N4: nat] :
            ( ( Y
              = ( suc @ N4 ) )
            & ( X
              = ( extended_enat2 @ N4 ) ) ) ) ) ).

% eSuc_enat_iff
thf(fact_9381_eSuc__enat,axiom,
    ! [N: nat] :
      ( ( extended_eSuc @ ( extended_enat2 @ N ) )
      = ( extended_enat2 @ ( suc @ N ) ) ) ).

% eSuc_enat
thf(fact_9382_mult__eSuc__right,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat @ M2 @ ( extended_eSuc @ N ) )
      = ( plus_p3455044024723400733d_enat @ M2 @ ( times_7803423173614009249d_enat @ M2 @ N ) ) ) ).

% mult_eSuc_right
thf(fact_9383_mult__eSuc,axiom,
    ! [M2: extended_enat,N: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( extended_eSuc @ M2 ) @ N )
      = ( plus_p3455044024723400733d_enat @ N @ ( times_7803423173614009249d_enat @ M2 @ N ) ) ) ).

% mult_eSuc
thf(fact_9384_less__than__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ less_than )
      = ( ord_less_nat @ X @ Y ) ) ).

% less_than_iff
thf(fact_9385_Real_Opositive_Orsp,axiom,
    ( bNF_re728719798268516973at_o_o @ realrel
    @ ^ [Y5: $o,Z2: $o] : Y5 = Z2
    @ ^ [X7: nat > rat] :
      ? [R: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R @ ( X7 @ N4 ) ) ) )
    @ ^ [X7: nat > rat] :
      ? [R: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R @ ( X7 @ N4 ) ) ) ) ) ).

% Real.positive.rsp
thf(fact_9386_Gcd__nat__set__eq__fold,axiom,
    ! [Xs2: list_nat] :
      ( ( gcd_Gcd_nat @ ( set_nat2 @ Xs2 ) )
      = ( fold_nat_nat @ gcd_gcd_nat @ Xs2 @ zero_zero_nat ) ) ).

% Gcd_nat_set_eq_fold
thf(fact_9387_times__real_Orsp,axiom,
    ( bNF_re1962705104956426057at_rat @ realrel @ ( bNF_re895249473297799549at_rat @ realrel @ realrel )
    @ ^ [X7: nat > rat,Y9: nat > rat,N4: nat] : ( times_times_rat @ ( X7 @ N4 ) @ ( Y9 @ N4 ) )
    @ ^ [X7: nat > rat,Y9: nat > rat,N4: nat] : ( times_times_rat @ ( X7 @ N4 ) @ ( Y9 @ N4 ) ) ) ).

% times_real.rsp
thf(fact_9388_Real_Opositive_Otransfer,axiom,
    ( bNF_re4297313714947099218al_o_o @ pcr_real
    @ ^ [Y5: $o,Z2: $o] : Y5 = Z2
    @ ^ [X7: nat > rat] :
      ? [R: rat] :
        ( ( ord_less_rat @ zero_zero_rat @ R )
        & ? [K3: nat] :
          ! [N4: nat] :
            ( ( ord_less_eq_nat @ K3 @ N4 )
           => ( ord_less_rat @ R @ ( X7 @ N4 ) ) ) )
    @ positive2 ) ).

% Real.positive.transfer
thf(fact_9389_Real_Opositive__mult,axiom,
    ! [X: real,Y: real] :
      ( ( positive2 @ X )
     => ( ( positive2 @ Y )
       => ( positive2 @ ( times_times_real @ X @ Y ) ) ) ) ).

% Real.positive_mult
thf(fact_9390_times__real_Otransfer,axiom,
    ( bNF_re4695409256820837752l_real @ pcr_real @ ( bNF_re3023117138289059399t_real @ pcr_real @ pcr_real )
    @ ^ [X7: nat > rat,Y9: nat > rat,N4: nat] : ( times_times_rat @ ( X7 @ N4 ) @ ( Y9 @ N4 ) )
    @ times_times_real ) ).

% times_real.transfer
thf(fact_9391_Real_Opositive_Orep__eq,axiom,
    ( positive2
    = ( ^ [X4: real] :
        ? [R: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R )
          & ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ R @ ( rep_real @ X4 @ N4 ) ) ) ) ) ) ).

% Real.positive.rep_eq
thf(fact_9392_Real_Opositive__def,axiom,
    ( positive2
    = ( map_fu1856342031159181835at_o_o @ rep_real @ id_o
      @ ^ [X7: nat > rat] :
        ? [R: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R )
          & ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ R @ ( X7 @ N4 ) ) ) ) ) ) ).

% Real.positive_def
thf(fact_9393_Real_Opositive_Oabs__eq,axiom,
    ! [X: nat > rat] :
      ( ( realrel @ X @ X )
     => ( ( positive2 @ ( real2 @ X ) )
        = ( ? [R: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R )
              & ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ R @ ( X @ N4 ) ) ) ) ) ) ) ).

% Real.positive.abs_eq
thf(fact_9394_times__real_Oabs__eq,axiom,
    ! [Xa2: nat > rat,X: nat > rat] :
      ( ( realrel @ Xa2 @ Xa2 )
     => ( ( realrel @ X @ X )
       => ( ( times_times_real @ ( real2 @ Xa2 ) @ ( real2 @ X ) )
          = ( real2
            @ ^ [N4: nat] : ( times_times_rat @ ( Xa2 @ N4 ) @ ( X @ N4 ) ) ) ) ) ) ).

% times_real.abs_eq
thf(fact_9395_le__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( ord_less_eq_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( ! [R: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ R )
               => ? [K3: nat] :
                  ! [N4: nat] :
                    ( ( ord_less_eq_nat @ K3 @ N4 )
                   => ( ord_less_eq_rat @ ( X8 @ N4 ) @ ( plus_plus_rat @ ( Y7 @ N4 ) @ R ) ) ) ) ) ) ) ) ).

% le_Real
thf(fact_9396_not__positive__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
        = ( ! [R: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R )
             => ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_eq_rat @ ( X8 @ N4 ) @ R ) ) ) ) ) ) ).

% not_positive_Real
thf(fact_9397_cauchy__mult,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( cauchy
          @ ^ [N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).

% cauchy_mult
thf(fact_9398_mult__Real,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( cauchy @ Y7 )
       => ( ( times_times_real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
          = ( real2
            @ ^ [N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ) ).

% mult_Real
thf(fact_9399_cauchyD,axiom,
    ! [X8: nat > rat,R3: rat] :
      ( ( cauchy @ X8 )
     => ( ( ord_less_rat @ zero_zero_rat @ R3 )
       => ? [K: nat] :
          ! [M3: nat] :
            ( ( ord_less_eq_nat @ K @ M3 )
           => ! [N8: nat] :
                ( ( ord_less_eq_nat @ K @ N8 )
               => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M3 ) @ ( X8 @ N8 ) ) ) @ R3 ) ) ) ) ) ).

% cauchyD
thf(fact_9400_cauchyI,axiom,
    ! [X8: nat > rat] :
      ( ! [R4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R4 )
         => ? [K4: nat] :
            ! [M: nat] :
              ( ( ord_less_eq_nat @ K4 @ M )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ K4 @ N3 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X8 @ M ) @ ( X8 @ N3 ) ) ) @ R4 ) ) ) )
     => ( cauchy @ X8 ) ) ).

% cauchyI
thf(fact_9401_cauchy__def,axiom,
    ( cauchy
    = ( ^ [X7: nat > rat] :
        ! [R: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R )
         => ? [K3: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ K3 @ M6 )
             => ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( X7 @ M6 ) @ ( X7 @ N4 ) ) ) @ R ) ) ) ) ) ) ).

% cauchy_def
thf(fact_9402_positive__Real,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ( positive2 @ ( real2 @ X8 ) )
        = ( ? [R: rat] :
              ( ( ord_less_rat @ zero_zero_rat @ R )
              & ? [K3: nat] :
                ! [N4: nat] :
                  ( ( ord_less_eq_nat @ K3 @ N4 )
                 => ( ord_less_rat @ R @ ( X8 @ N4 ) ) ) ) ) ) ) ).

% positive_Real
thf(fact_9403_cauchy__not__vanishes__cases,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ? [B3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B3 )
            & ? [K: nat] :
                ( ! [N8: nat] :
                    ( ( ord_less_eq_nat @ K @ N8 )
                   => ( ord_less_rat @ B3 @ ( uminus_uminus_rat @ ( X8 @ N8 ) ) ) )
                | ! [N8: nat] :
                    ( ( ord_less_eq_nat @ K @ N8 )
                   => ( ord_less_rat @ B3 @ ( X8 @ N8 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes_cases
thf(fact_9404_cauchy__not__vanishes,axiom,
    ! [X8: nat > rat] :
      ( ( cauchy @ X8 )
     => ( ~ ( vanishes @ X8 )
       => ? [B3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ B3 )
            & ? [K: nat] :
              ! [N8: nat] :
                ( ( ord_less_eq_nat @ K @ N8 )
               => ( ord_less_rat @ B3 @ ( abs_abs_rat @ ( X8 @ N8 ) ) ) ) ) ) ) ).

% cauchy_not_vanishes
thf(fact_9405_vanishesD,axiom,
    ! [X8: nat > rat,R3: rat] :
      ( ( vanishes @ X8 )
     => ( ( ord_less_rat @ zero_zero_rat @ R3 )
       => ? [K: nat] :
          ! [N8: nat] :
            ( ( ord_less_eq_nat @ K @ N8 )
           => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N8 ) ) @ R3 ) ) ) ) ).

% vanishesD
thf(fact_9406_vanishesI,axiom,
    ! [X8: nat > rat] :
      ( ! [R4: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R4 )
         => ? [K4: nat] :
            ! [N3: nat] :
              ( ( ord_less_eq_nat @ K4 @ N3 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ R4 ) ) )
     => ( vanishes @ X8 ) ) ).

% vanishesI
thf(fact_9407_vanishes__def,axiom,
    ( vanishes
    = ( ^ [X7: nat > rat] :
        ! [R: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ R )
         => ? [K3: nat] :
            ! [N4: nat] :
              ( ( ord_less_eq_nat @ K3 @ N4 )
             => ( ord_less_rat @ ( abs_abs_rat @ ( X7 @ N4 ) ) @ R ) ) ) ) ) ).

% vanishes_def
thf(fact_9408_vanishes__mult__bounded,axiom,
    ! [X8: nat > rat,Y7: nat > rat] :
      ( ? [A6: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ A6 )
          & ! [N3: nat] : ( ord_less_rat @ ( abs_abs_rat @ ( X8 @ N3 ) ) @ A6 ) )
     => ( ( vanishes @ Y7 )
       => ( vanishes
          @ ^ [N4: nat] : ( times_times_rat @ ( X8 @ N4 ) @ ( Y7 @ N4 ) ) ) ) ) ).

% vanishes_mult_bounded
thf(fact_9409_surj__int__decode,axiom,
    ( ( image_nat_int @ nat_int_decode @ top_top_set_nat )
    = top_top_set_int ) ).

% surj_int_decode
thf(fact_9410_bij__int__decode,axiom,
    bij_betw_nat_int @ nat_int_decode @ top_top_set_nat @ top_top_set_int ).

% bij_int_decode
thf(fact_9411_inj__int__decode,axiom,
    ! [A4: set_nat] : ( inj_on_nat_int @ nat_int_decode @ A4 ) ).

% inj_int_decode
thf(fact_9412_int__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_int_decode @ X )
        = ( nat_int_decode @ Y ) )
      = ( X = Y ) ) ).

% int_decode_eq
thf(fact_9413_nat__to__rat__surj__def,axiom,
    ( nat_to_rat_surj
    = ( ^ [N4: nat] :
          ( produc6207742614233964070at_rat
          @ ^ [A5: nat,B4: nat] : ( fract @ ( nat_int_decode @ A5 ) @ ( nat_int_decode @ B4 ) )
          @ ( nat_prod_decode @ N4 ) ) ) ) ).

% nat_to_rat_surj_def
thf(fact_9414_bij__int__encode,axiom,
    bij_betw_int_nat @ nat_int_encode @ top_top_set_int @ top_top_set_nat ).

% bij_int_encode
thf(fact_9415_int__encode__inverse,axiom,
    ! [X: int] :
      ( ( nat_int_decode @ ( nat_int_encode @ X ) )
      = X ) ).

% int_encode_inverse
thf(fact_9416_int__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_int_encode @ ( nat_int_decode @ N ) )
      = N ) ).

% int_decode_inverse
thf(fact_9417_inj__int__encode,axiom,
    ! [A4: set_int] : ( inj_on_int_nat @ nat_int_encode @ A4 ) ).

% inj_int_encode
thf(fact_9418_int__encode__eq,axiom,
    ! [X: int,Y: int] :
      ( ( ( nat_int_encode @ X )
        = ( nat_int_encode @ Y ) )
      = ( X = Y ) ) ).

% int_encode_eq
thf(fact_9419_surj__int__encode,axiom,
    ( ( image_int_nat @ nat_int_encode @ top_top_set_int )
    = top_top_set_nat ) ).

% surj_int_encode
thf(fact_9420_linear__scale__real,axiom,
    ! [F2: real > real,R3: real,B: real] :
      ( ( real_V4572627801940501177l_real @ F2 )
     => ( ( F2 @ ( times_times_real @ R3 @ B ) )
        = ( times_times_real @ R3 @ ( F2 @ B ) ) ) ) ).

% linear_scale_real
thf(fact_9421_real__linearD,axiom,
    ! [F2: real > real] :
      ( ( real_V4572627801940501177l_real @ F2 )
     => ~ ! [C: real] :
            ( F2
           != ( times_times_real @ C ) ) ) ).

% real_linearD
thf(fact_9422_natLeq__Partial__order,axiom,
    order_5251275573222108571on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Partial_order
thf(fact_9423_natLeq__Preorder,axiom,
    order_4861654808422542329on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Preorder
thf(fact_9424_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X4: nat,Y4: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X4 @ V3 ) @ ( plus_plus_nat @ U2 @ Y4 ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9425_int__encode__def,axiom,
    ( nat_int_encode
    = ( ^ [I: int] : ( nat_sum_encode @ ( if_Sum_sum_nat_nat @ ( ord_less_eq_int @ zero_zero_int @ I ) @ ( sum_Inl_nat_nat @ ( nat2 @ I ) ) @ ( sum_Inr_nat_nat @ ( nat2 @ ( minus_minus_int @ ( uminus_uminus_int @ I ) @ one_one_int ) ) ) ) ) ) ) ).

% int_encode_def
thf(fact_9426_surj__sum__encode,axiom,
    ( ( image_1320371278474632150at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat )
    = top_top_set_nat ) ).

% surj_sum_encode
thf(fact_9427_inj__sum__encode,axiom,
    ! [A4: set_Sum_sum_nat_nat] : ( inj_on6343450744447823682at_nat @ nat_sum_encode @ A4 ) ).

% inj_sum_encode
thf(fact_9428_sum__encode__eq,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
      ( ( ( nat_sum_encode @ X )
        = ( nat_sum_encode @ Y ) )
      = ( X = Y ) ) ).

% sum_encode_eq
thf(fact_9429_bij__sum__encode,axiom,
    bij_be5432664580149595207at_nat @ nat_sum_encode @ top_to6661820994512907621at_nat @ top_top_set_nat ).

% bij_sum_encode
thf(fact_9430_le__sum__encode__Inl,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ ( nat_sum_encode @ ( sum_Inl_nat_nat @ Y ) ) ) ) ).

% le_sum_encode_Inl
thf(fact_9431_le__sum__encode__Inr,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ ( nat_sum_encode @ ( sum_Inr_nat_nat @ Y ) ) ) ) ).

% le_sum_encode_Inr
thf(fact_9432_sum__decode__def,axiom,
    ( nat_sum_decode
    = ( ^ [N4: nat] : ( if_Sum_sum_nat_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N4 ) @ ( sum_Inl_nat_nat @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( sum_Inr_nat_nat @ ( divide_divide_nat @ N4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sum_decode_def
thf(fact_9433_sum__encode__inverse,axiom,
    ! [X: sum_sum_nat_nat] :
      ( ( nat_sum_decode @ ( nat_sum_encode @ X ) )
      = X ) ).

% sum_encode_inverse
thf(fact_9434_sum__decode__inverse,axiom,
    ! [N: nat] :
      ( ( nat_sum_encode @ ( nat_sum_decode @ N ) )
      = N ) ).

% sum_decode_inverse
thf(fact_9435_bij__sum__decode,axiom,
    bij_be4790990086886966983at_nat @ nat_sum_decode @ top_top_set_nat @ top_to6661820994512907621at_nat ).

% bij_sum_decode
thf(fact_9436_sum__decode__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( nat_sum_decode @ X )
        = ( nat_sum_decode @ Y ) )
      = ( X = Y ) ) ).

% sum_decode_eq
thf(fact_9437_inj__sum__decode,axiom,
    ! [A4: set_nat] : ( inj_on5701776251185195458at_nat @ nat_sum_decode @ A4 ) ).

% inj_sum_decode
thf(fact_9438_surj__sum__decode,axiom,
    ( ( image_678696785212003926at_nat @ nat_sum_decode @ top_top_set_nat )
    = top_to6661820994512907621at_nat ) ).

% surj_sum_decode
thf(fact_9439_nth__item_Opinduct,axiom,
    ! [A0: nat,P2: nat > $o] :
      ( ( accp_nat @ nth_item_rel @ A0 )
     => ( ( ( accp_nat @ nth_item_rel @ zero_zero_nat )
         => ( P2 @ zero_zero_nat ) )
       => ( ! [N3: nat] :
              ( ( accp_nat @ nth_item_rel @ ( suc @ N3 ) )
             => ( ! [A6: nat,Aa: nat] :
                    ( ( ( nat_sum_decode @ N3 )
                      = ( sum_Inl_nat_nat @ A6 ) )
                   => ( ( ( nat_sum_decode @ A6 )
                        = ( sum_Inl_nat_nat @ Aa ) )
                     => ( P2 @ Aa ) ) )
               => ( ! [A6: nat,B6: nat] :
                      ( ( ( nat_sum_decode @ N3 )
                        = ( sum_Inl_nat_nat @ A6 ) )
                     => ( ( ( nat_sum_decode @ A6 )
                          = ( sum_Inr_nat_nat @ B6 ) )
                       => ( P2 @ B6 ) ) )
                 => ( ! [B6: nat,Ba: nat,X5: nat,Y6: nat] :
                        ( ( ( nat_sum_decode @ N3 )
                          = ( sum_Inr_nat_nat @ B6 ) )
                       => ( ( ( nat_sum_decode @ B6 )
                            = ( sum_Inr_nat_nat @ Ba ) )
                         => ( ( ( product_Pair_nat_nat @ X5 @ Y6 )
                              = ( nat_prod_decode @ Ba ) )
                           => ( P2 @ X5 ) ) ) )
                   => ( ! [B6: nat,Ba: nat,X5: nat,Y6: nat] :
                          ( ( ( nat_sum_decode @ N3 )
                            = ( sum_Inr_nat_nat @ B6 ) )
                         => ( ( ( nat_sum_decode @ B6 )
                              = ( sum_Inr_nat_nat @ Ba ) )
                           => ( ( ( product_Pair_nat_nat @ X5 @ Y6 )
                                = ( nat_prod_decode @ Ba ) )
                             => ( P2 @ Y6 ) ) ) )
                     => ( P2 @ ( suc @ N3 ) ) ) ) ) ) )
         => ( P2 @ A0 ) ) ) ) ).

% nth_item.pinduct
thf(fact_9440_int__decode__def,axiom,
    ( nat_int_decode
    = ( ^ [N4: nat] :
          ( sum_ca7763040182479039464nt_nat @ semiri1314217659103216013at_int
          @ ^ [B4: nat] : ( minus_minus_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ B4 ) ) @ one_one_int )
          @ ( nat_sum_decode @ N4 ) ) ) ) ).

% int_decode_def
thf(fact_9441_sum__encode__def,axiom,
    ( nat_sum_encode
    = ( sum_ca6763686470577984908at_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      @ ^ [B4: nat] : ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) ) ) ) ).

% sum_encode_def
thf(fact_9442_enat__def,axiom,
    ( extended_enat2
    = ( ^ [N4: nat] : ( extended_Abs_enat @ ( some_nat @ N4 ) ) ) ) ).

% enat_def
thf(fact_9443_Bseq__monoseq__convergent_H__dec,axiom,
    ! [F2: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N4: nat] : ( F2 @ ( plus_plus_nat @ N4 @ M5 ) )
        @ at_top_nat )
     => ( ! [M: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ M )
           => ( ( ord_less_eq_nat @ M @ N3 )
             => ( ord_less_eq_real @ ( F2 @ N3 ) @ ( F2 @ M ) ) ) )
       => ( topolo7531315842566124627t_real @ F2 ) ) ) ).

% Bseq_monoseq_convergent'_dec
thf(fact_9444_Bseq__monoseq__convergent_H__inc,axiom,
    ! [F2: nat > real,M5: nat] :
      ( ( bfun_nat_real
        @ ^ [N4: nat] : ( F2 @ ( plus_plus_nat @ N4 @ M5 ) )
        @ at_top_nat )
     => ( ! [M: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M5 @ M )
           => ( ( ord_less_eq_nat @ M @ N3 )
             => ( ord_less_eq_real @ ( F2 @ M ) @ ( F2 @ N3 ) ) ) )
       => ( topolo7531315842566124627t_real @ F2 ) ) ) ).

% Bseq_monoseq_convergent'_inc
thf(fact_9445_Bseq__mono__convergent,axiom,
    ! [X8: nat > real] :
      ( ( bfun_nat_real @ X8 @ at_top_nat )
     => ( ! [M: nat,N3: nat] :
            ( ( ord_less_eq_nat @ M @ N3 )
           => ( ord_less_eq_real @ ( X8 @ M ) @ ( X8 @ N3 ) ) )
       => ( topolo7531315842566124627t_real @ X8 ) ) ) ).

% Bseq_mono_convergent
thf(fact_9446_lcm__altdef__int,axiom,
    ( gcd_lcm_int
    = ( ^ [A5: int,B4: int] : ( divide_divide_int @ ( times_times_int @ ( abs_abs_int @ A5 ) @ ( abs_abs_int @ B4 ) ) @ ( gcd_gcd_int @ A5 @ B4 ) ) ) ) ).

% lcm_altdef_int
thf(fact_9447_prod__gcd__lcm__int,axiom,
    ! [M2: int,N: int] :
      ( ( times_times_int @ ( abs_abs_int @ M2 ) @ ( abs_abs_int @ N ) )
      = ( times_times_int @ ( gcd_gcd_int @ M2 @ N ) @ ( gcd_lcm_int @ M2 @ N ) ) ) ).

% prod_gcd_lcm_int
thf(fact_9448_lcm__0__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        | ( N = zero_zero_nat ) ) ) ).

% lcm_0_iff_nat
thf(fact_9449_lcm__1__iff__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( gcd_lcm_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( M2
          = ( suc @ zero_zero_nat ) )
        & ( N
          = ( suc @ zero_zero_nat ) ) ) ) ).

% lcm_1_iff_nat
thf(fact_9450_prod__gcd__lcm__nat,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N4: nat] : ( times_times_nat @ ( gcd_gcd_nat @ M6 @ N4 ) @ ( gcd_lcm_nat @ M6 @ N4 ) ) ) ) ).

% prod_gcd_lcm_nat
thf(fact_9451_lcm__pos__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N )
       => ( ord_less_nat @ zero_zero_nat @ ( gcd_lcm_nat @ M2 @ N ) ) ) ) ).

% lcm_pos_nat
thf(fact_9452_lcm__code__integer,axiom,
    ( gcd_lcm_Code_integer
    = ( ^ [A5: code_integer,B4: code_integer] : ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A5 ) @ ( abs_abs_Code_integer @ B4 ) ) @ ( gcd_gcd_Code_integer @ A5 @ B4 ) ) ) ) ).

% lcm_code_integer
thf(fact_9453_lcm__nat__def,axiom,
    ( gcd_lcm_nat
    = ( ^ [X4: nat,Y4: nat] : ( divide_divide_nat @ ( times_times_nat @ X4 @ Y4 ) @ ( gcd_gcd_nat @ X4 @ Y4 ) ) ) ) ).

% lcm_nat_def
thf(fact_9454_Lcm__eq__Max__nat,axiom,
    ! [M5: set_nat] :
      ( ( finite_finite_nat @ M5 )
     => ( ( M5 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M5 )
         => ( ! [M: nat,N3: nat] :
                ( ( member_nat @ M @ M5 )
               => ( ( member_nat @ N3 @ M5 )
                 => ( member_nat @ ( gcd_lcm_nat @ M @ N3 ) @ M5 ) ) )
           => ( ( gcd_Lcm_nat @ M5 )
              = ( lattic8265883725875713057ax_nat @ M5 ) ) ) ) ) ) ).

% Lcm_eq_Max_nat
thf(fact_9455_Lcm__eq__0__I__nat,axiom,
    ! [A4: set_nat] :
      ( ( member_nat @ zero_zero_nat @ A4 )
     => ( ( gcd_Lcm_nat @ A4 )
        = zero_zero_nat ) ) ).

% Lcm_eq_0_I_nat
thf(fact_9456_Lcm__0__iff__nat,axiom,
    ! [A4: set_nat] :
      ( ( finite_finite_nat @ A4 )
     => ( ( ( gcd_Lcm_nat @ A4 )
          = zero_zero_nat )
        = ( member_nat @ zero_zero_nat @ A4 ) ) ) ).

% Lcm_0_iff_nat
thf(fact_9457_Lcm__nat__infinite,axiom,
    ! [M5: set_nat] :
      ( ~ ( finite_finite_nat @ M5 )
     => ( ( gcd_Lcm_nat @ M5 )
        = zero_zero_nat ) ) ).

% Lcm_nat_infinite
thf(fact_9458_Lcm__nat__def,axiom,
    ( gcd_Lcm_nat
    = ( ^ [M9: set_nat] : ( if_nat @ ( finite_finite_nat @ M9 ) @ ( lattic7826324295020591184_F_nat @ gcd_lcm_nat @ one_one_nat @ M9 ) @ zero_zero_nat ) ) ) ).

% Lcm_nat_def
thf(fact_9459_unit__factor__simps_I1_J,axiom,
    ( ( unit_f2748546683901255202or_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% unit_factor_simps(1)
thf(fact_9460_unit__factor__simps_I2_J,axiom,
    ! [N: nat] :
      ( ( unit_f2748546683901255202or_nat @ ( suc @ N ) )
      = one_one_nat ) ).

% unit_factor_simps(2)
thf(fact_9461_unit__factor__nat__def,axiom,
    ( unit_f2748546683901255202or_nat
    = ( ^ [N4: nat] : ( if_nat @ ( N4 = zero_zero_nat ) @ zero_zero_nat @ one_one_nat ) ) ) ).

% unit_factor_nat_def
thf(fact_9462_times__num__def,axiom,
    ( times_times_num
    = ( ^ [M6: num,N4: num] : ( num_of_nat @ ( times_times_nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).

% times_num_def
thf(fact_9463_nat__of__num__code_I2_J,axiom,
    ! [N: num] :
      ( ( nat_of_num @ ( bit0 @ N ) )
      = ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).

% nat_of_num_code(2)
thf(fact_9464_less__eq__num__def,axiom,
    ( ord_less_eq_num
    = ( ^ [M6: num,N4: num] : ( ord_less_eq_nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N4 ) ) ) ) ).

% less_eq_num_def
thf(fact_9465_less__num__def,axiom,
    ( ord_less_num
    = ( ^ [M6: num,N4: num] : ( ord_less_nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N4 ) ) ) ) ).

% less_num_def
thf(fact_9466_nat__of__num__inc,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( inc @ X ) )
      = ( suc @ ( nat_of_num @ X ) ) ) ).

% nat_of_num_inc
thf(fact_9467_nat__of__num__mult,axiom,
    ! [X: num,Y: num] :
      ( ( nat_of_num @ ( times_times_num @ X @ Y ) )
      = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).

% nat_of_num_mult
thf(fact_9468_nat__of__num__numeral,axiom,
    nat_of_num = numeral_numeral_nat ).

% nat_of_num_numeral
thf(fact_9469_num__eq__iff,axiom,
    ( ( ^ [Y5: num,Z2: num] : Y5 = Z2 )
    = ( ^ [X4: num,Y4: num] :
          ( ( nat_of_num @ X4 )
          = ( nat_of_num @ Y4 ) ) ) ) ).

% num_eq_iff
thf(fact_9470_nat__of__num__inverse,axiom,
    ! [X: num] :
      ( ( num_of_nat @ ( nat_of_num @ X ) )
      = X ) ).

% nat_of_num_inverse
thf(fact_9471_nat__of__num_Osimps_I2_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit0 @ X ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).

% nat_of_num.simps(2)
thf(fact_9472_nat__of__num__pos,axiom,
    ! [X: num] : ( ord_less_nat @ zero_zero_nat @ ( nat_of_num @ X ) ) ).

% nat_of_num_pos
thf(fact_9473_nat__of__num__neq__0,axiom,
    ! [X: num] :
      ( ( nat_of_num @ X )
     != zero_zero_nat ) ).

% nat_of_num_neq_0
thf(fact_9474_nat__of__num__code_I1_J,axiom,
    ( ( nat_of_num @ one )
    = one_one_nat ) ).

% nat_of_num_code(1)
thf(fact_9475_nat__of__num__add,axiom,
    ! [X: num,Y: num] :
      ( ( nat_of_num @ ( plus_plus_num @ X @ Y ) )
      = ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y ) ) ) ).

% nat_of_num_add
thf(fact_9476_nat__of__num__sqr,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( sqr @ X ) )
      = ( times_times_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).

% nat_of_num_sqr
thf(fact_9477_nat__of__num_Osimps_I1_J,axiom,
    ( ( nat_of_num @ one )
    = ( suc @ zero_zero_nat ) ) ).

% nat_of_num.simps(1)
thf(fact_9478_nat__of__num_Osimps_I3_J,axiom,
    ! [X: num] :
      ( ( nat_of_num @ ( bit1 @ X ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).

% nat_of_num.simps(3)
thf(fact_9479_num__of__nat__inverse,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( nat_of_num @ ( num_of_nat @ N ) )
        = N ) ) ).

% num_of_nat_inverse
thf(fact_9480_nat__of__num__code_I3_J,axiom,
    ! [N: num] :
      ( ( nat_of_num @ ( bit1 @ N ) )
      = ( suc @ ( plus_plus_nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).

% nat_of_num_code(3)
thf(fact_9481_plus__num__def,axiom,
    ( plus_plus_num
    = ( ^ [M6: num,N4: num] : ( num_of_nat @ ( plus_plus_nat @ ( nat_of_num @ M6 ) @ ( nat_of_num @ N4 ) ) ) ) ) ).

% plus_num_def
thf(fact_9482_natLeq__trans,axiom,
    trans_nat @ bNF_Ca8665028551170535155natLeq ).

% natLeq_trans
thf(fact_9483_natLeq__antisym,axiom,
    antisym_nat @ bNF_Ca8665028551170535155natLeq ).

% natLeq_antisym
thf(fact_9484_VEBT__internal_Olesseq_Oelims_I3_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ~ ( vEBT_VEBT_lesseq @ X @ Xa2 )
     => ~ ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X @ Xa2 ) ) ).

% VEBT_internal.lesseq.elims(3)
thf(fact_9485_VEBT__internal_Olesseq_Oelims_I2_J,axiom,
    ! [X: option_nat,Xa2: option_nat] :
      ( ( vEBT_VEBT_lesseq @ X @ Xa2 )
     => ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X @ Xa2 ) ) ).

% VEBT_internal.lesseq.elims(2)
thf(fact_9486_VEBT__internal_Olesseq_Oelims_I1_J,axiom,
    ! [X: option_nat,Xa2: option_nat,Y: $o] :
      ( ( ( vEBT_VEBT_lesseq @ X @ Xa2 )
        = Y )
     => ( Y
        = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat @ X @ Xa2 ) ) ) ).

% VEBT_internal.lesseq.elims(1)
thf(fact_9487_VEBT__internal_Olesseq_Osimps,axiom,
    ( vEBT_VEBT_lesseq
    = ( vEBT_V2881884560877996034ft_nat @ ord_less_eq_nat ) ) ).

% VEBT_internal.lesseq.simps
thf(fact_9488_integer__of__nat__0,axiom,
    ( ( code_integer_of_nat @ zero_zero_nat )
    = zero_z3403309356797280102nteger ) ).

% integer_of_nat_0
thf(fact_9489_natural__decr,axiom,
    ! [N: code_natural] :
      ( ( N != zero_z2226904508553997617atural )
     => ( ord_less_nat @ ( minus_minus_nat @ ( code_nat_of_natural @ N ) @ ( suc @ zero_zero_nat ) ) @ ( code_nat_of_natural @ N ) ) ) ).

% natural_decr
thf(fact_9490_times__natural_Orep__eq,axiom,
    ! [X: code_natural,Xa2: code_natural] :
      ( ( code_nat_of_natural @ ( times_2397367101498566445atural @ X @ Xa2 ) )
      = ( times_times_nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa2 ) ) ) ).

% times_natural.rep_eq
thf(fact_9491_zero__natural_Orep__eq,axiom,
    ( ( code_nat_of_natural @ zero_z2226904508553997617atural )
    = zero_zero_nat ) ).

% zero_natural.rep_eq
thf(fact_9492_less__eq__natural_Orep__eq,axiom,
    ( ord_le1926595141338095240atural
    = ( ^ [X4: code_natural,Xa3: code_natural] : ( ord_less_eq_nat @ ( code_nat_of_natural @ X4 ) @ ( code_nat_of_natural @ Xa3 ) ) ) ) ).

% less_eq_natural.rep_eq
thf(fact_9493_next_Osimps,axiom,
    ! [V2: code_natural,W2: code_natural] :
      ( ( next @ ( produc3574140220909816553atural @ V2 @ W2 ) )
      = ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_p4538020629002901425atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ one_one_Code_natural ) ) @ one_one_Code_natural ) @ ( produc3574140220909816553atural @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ V2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( modulo8411746178871703098atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_2397367101498566445atural @ ( divide5121882707175180666atural @ W2 @ ( numera5444537566228673987atural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numera5444537566228673987atural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% next.simps
thf(fact_9494_Random_Orange__def,axiom,
    ( range
    = ( ^ [K3: code_natural] :
          ( produc5538323210962509403atural
          @ ( iterat8892046348760725948atural @ ( log @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
            @ ^ [L2: code_natural] :
                ( produc5538323210962509403atural @ next
                @ ^ [V3: code_natural] : ( produc6639722614265839536atural @ ( plus_p4538020629002901425atural @ V3 @ ( times_2397367101498566445atural @ L2 @ ( numera5444537566228673987atural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
            @ one_one_Code_natural )
          @ ^ [V3: code_natural] : ( produc6639722614265839536atural @ ( modulo8411746178871703098atural @ V3 @ K3 ) ) ) ) ) ).

% Random.range_def
thf(fact_9495_Suc_Orep__eq,axiom,
    ! [X: code_natural] :
      ( ( code_nat_of_natural @ ( code_Suc @ X ) )
      = ( suc @ ( code_nat_of_natural @ X ) ) ) ).

% Suc.rep_eq
thf(fact_9496_Suc_Oabs__eq,axiom,
    ! [X: nat] :
      ( ( code_Suc @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( suc @ X ) ) ) ).

% Suc.abs_eq
thf(fact_9497_less__eq__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( ( ord_le1926595141338095240atural @ ( code_natural_of_nat @ Xa2 ) @ ( code_natural_of_nat @ X ) )
      = ( ord_less_eq_nat @ Xa2 @ X ) ) ).

% less_eq_natural.abs_eq
thf(fact_9498_times__natural_Oabs__eq,axiom,
    ! [Xa2: nat,X: nat] :
      ( ( times_2397367101498566445atural @ ( code_natural_of_nat @ Xa2 ) @ ( code_natural_of_nat @ X ) )
      = ( code_natural_of_nat @ ( times_times_nat @ Xa2 @ X ) ) ) ).

% times_natural.abs_eq
thf(fact_9499_less__nat__rel,axiom,
    ( ord_less_nat
    = ( transi2163837189807498211lp_nat
      @ ^ [M6: nat,N4: nat] :
          ( N4
          = ( suc @ M6 ) ) ) ) ).

% less_nat_rel
thf(fact_9500_zero__natural__def,axiom,
    ( zero_z2226904508553997617atural
    = ( code_natural_of_nat @ zero_zero_nat ) ) ).

% zero_natural_def
thf(fact_9501_Code__Numeral_OSuc__def,axiom,
    ( code_Suc
    = ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).

% Code_Numeral.Suc_def
thf(fact_9502_times__natural__def,axiom,
    ( times_2397367101498566445atural
    = ( map_fu6549440983881763648atural @ code_nat_of_natural @ ( map_fu1239815594074539274atural @ code_nat_of_natural @ code_natural_of_nat ) @ times_times_nat ) ) ).

% times_natural_def
thf(fact_9503_typerep_Osize__neq,axiom,
    ! [X: typerep] :
      ( ( size_size_typerep @ X )
     != zero_zero_nat ) ).

% typerep.size_neq
thf(fact_9504_typerep_Osize_I2_J,axiom,
    ! [X1: literal,X2: list_typerep] :
      ( ( size_size_typerep @ ( typerep2 @ X1 @ X2 ) )
      = ( plus_plus_nat @ ( size_list_typerep @ size_size_typerep @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% typerep.size(2)
thf(fact_9505_typerep_Osize__gen,axiom,
    ! [X1: literal,X2: list_typerep] :
      ( ( size_typerep @ ( typerep2 @ X1 @ X2 ) )
      = ( plus_plus_nat @ ( size_list_typerep @ size_typerep @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).

% typerep.size_gen
thf(fact_9506_max__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.semilattice_neutr_axioms
thf(fact_9507_gcd__nat_Osemilattice__neutr__axioms,axiom,
    semila9081495762789891438tr_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.semilattice_neutr_axioms
thf(fact_9508_real__times__code,axiom,
    ! [X: rat,Y: rat] :
      ( ( times_times_real @ ( ratreal @ X ) @ ( ratreal @ Y ) )
      = ( ratreal @ ( times_times_rat @ X @ Y ) ) ) ).

% real_times_code
thf(fact_9509_max__nat_Omonoid__axioms,axiom,
    monoid_nat @ ord_max_nat @ zero_zero_nat ).

% max_nat.monoid_axioms
thf(fact_9510_gcd__nat_Omonoid__axioms,axiom,
    monoid_nat @ gcd_gcd_nat @ zero_zero_nat ).

% gcd_nat.monoid_axioms
thf(fact_9511_natLeq__Card__order,axiom,
    bNF_Ca1281551314933786834on_nat @ ( field_nat @ bNF_Ca8665028551170535155natLeq ) @ bNF_Ca8665028551170535155natLeq ).

% natLeq_Card_order
thf(fact_9512_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_9513_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

% Helper facts (35)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X: int,Y: int] :
      ( ( if_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y: nat] :
      ( ( if_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X: num,Y: num] :
      ( ( if_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X: rat,Y: rat] :
      ( ( if_rat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X: real,Y: real] :
      ( ( if_real @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X: complex,Y: complex] :
      ( ( if_complex @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X: extended_enat,Y: extended_enat] :
      ( ( if_Extended_enat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Nat__Onat_J_T,axiom,
    ! [X: set_nat,Y: set_nat] :
      ( ( if_set_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__VEBT____Definitions__OVEBT_T,axiom,
    ! [X: vEBT_VEBT,Y: vEBT_VEBT] :
      ( ( if_VEBT_VEBT @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
    ! [X: option_nat,Y: option_nat] :
      ( ( if_option_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
      ( ( if_Sum_sum_nat_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: sum_sum_nat_nat,Y: sum_sum_nat_nat] :
      ( ( if_Sum_sum_nat_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( info
    = ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) ) ).

%------------------------------------------------------------------------------